Theory of Terahertz generation by Optical Rectification
M^SSACHURETT INSTI
by
OF TECHNOLOGY
Koustuban Ravi
SEP 2 5 2014
B.Eng., Nanyang Technological University (2009)
LIBRARIES
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2014
@2014 Massachusetts Institute of Technology. All rights reserved.
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Depkrtment of Electrical Engineering and Computer Science
August 29 2014
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Franz X.Krtner
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Thesis Supervisor
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Erich P.lppen
Elihu Thomson Professor of Electrical Engineering, Professor of Physics Emeritus
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Thesis Supervisor
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Leslie A. Kolodziejski
V %.Professor of Electrical Engineering
Chair, Department Committee on Graduate Students
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Theory of Terahertz Generation by Optical Rectification
by
Koustuban Ravi
Submitted to the Department of Electrical Engineering and Computer Science on August 29
2014, in partial fulfillment of the requirements for the degree of Master of Science
Abstract
Intense pulses of light with wavelengths approximately ten times smaller than microwave sources and a
hundred times larger than optical/near infra-red sources may be categorized as high-field Terahertz (THz)
sources. By virtue of their large electromagnetic field amplitudes and relatively long wavelengths, they
are uniquely amenable for electron acceleration, coherent X-ray generation and non-linear spectroscopy.
Intra-pulse difference frequency generation or optical rectification of ultrafast optical pump pulses in nonlinear crystals has emerged as the most efficient approach for high-field THz generation. Earlier
theoretical treatment of these systems had predicted conversion efficiencies as high as 10%, which
opened up the possibility of generating THz pulses with energies on the order of 10 milli joules on a
table-top. However, experimental demonstrations have achieved conversion efficiencies of only a few
percent which motivates a re-examination of the existing theory. In this thesis, we re-formulate the
problem by accounting for effects, previously not considered. These include: (i) the spatio-temporal
distortions of ultrafast pulses, (ii) the nonlinear coupled interaction of optical and THz radiation in two
spatial dimensions (2-D), (iii) self-phase modulation, (iv) stimulated Raman scattering and (v) crystal
geometry. The key finding is that THz generation necessarily leads to broadening of the optical pump
spectrum, resulting in a rapid spatio-temporal break-up of the pulse which limits further generation of
THz radiation. Due to this self-limiting mechanism, the predicted conversion efficiencies reduce
significantly in relation to earlier predictions, which is in line with experimental trends. Guidelines to
optimize conversion efficiency and their ramifications on spatial and spectral properties of THz radiation
are discussed. The predictions and analyses are supported by experiments. These findings direct future
work towards careful engineering of such systems to achieve optimal THz pulse properties and the
conception of approaches to circumvent the aforementioned self-limiting effects.
Thesis Supervisor: Franz X.Kdrtner
Title: Adjunct Professor of Electrical Engineering
Thesis Supervisor: Erich P.Ippen
Title: Elihu Thomson Professor of Electrical Engineering, Professor of Physics Emeritus
Thesis Supervisor
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Acknowledgment
I have had an incredible time at MIT over the course of the last two years. It is perhaps a cliche to state
that this place is special but one cannot emphasize sufficiently, how true this is. While much work needs
to be done to meet the institute's standards, it is befitting that I acknowledge those who have played a role
in my relative success (or more accurately, the lack of it) so far.
First, I would like to thank Prof. Sheel Aditya from Nanyang Technological University (NTU) for
introducing me to Electromagnetic Theory research, early in my sophomore year during my
undergraduate studies. His encouragement played an important role in spurring my scientific aspirations.
A key turning point in my career was working with Prof. Seng-Tiong Ho from Northwestern University.
He had a strong influence on my approach to solving problems, writing methodology and research
philosophy. The training I received from him was a key reason for me being able to conduct research
independently at graduate school. It would be fair to say that I probably would not have been at MIT if
not for his unwavering support.
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At MIT, I have had numerous individuals to look up to. Among staff in the Optics and Quantum
Electronics (OQE) group, I have had good role models in Billy Putnam, Donnie Keathley and Emilio
Nanni. The first papers I studied in the area of optics and photonics were those of Prof Ippen's and Prof
Hermann Haus. I am grateful to have Prof. Ippen as my co-advisor. Discussions with him have always
given me a fresh perspective on scientific and strategic fronts because of his clarity. On my very first day
of class at MIT, I was particularly impressed by Prof. Hagelstein's raw intellectual horsepower and hope
to match it at some point. I have had interesting conversations with every member of the OQE group
Damian Schimpf, Gregory Stein, Hung-Wen Chen, Jeffrey Moses, Kyung-Han Hong, Liang-Jie
Wong, Patrick Callahan and Siddharth Bhardwaj. In addition, I would like to thank Arya Fallahi,
Frederike Ahr, Sergio Carbajo and Xiaojun Wu from DESY in Hamburg for engaging in what
promises to be a fruitful collaboration. A special mention to my team mate on the terahertz project
Wenqian Huang for useful and important discussions on various fronts. Peter Krogen has been a great
friend and colleague.
I am fortunate to have Prof. Franz X. Kirtner as my advisor. Despite his numerous responsibilities, he
unbelievably makes time for discussions. He has given me ample freedom but also intervened at the right
times to ensure things stay on course. His broad scientific repertoire was a key reason which drew me to
the group but the most important pieces of knowledge that I have derived from Franz have been outside
the technical realm. His pleasant demeanor, patience, optimism and outstanding leadership skills are truly
worth emulating and have influenced me deeply.
Prof. Martin A. Schmidt, has promptly provided counsel whenever sought, in his capacity as my
graduate mentor.
I cannot thank my parents - Mohana Ravi and Ravi Gopalan enough for their herculean efforts in
raising, educating me and my brother - Ramgopal Ravi for exposing gaps in my perception and
entertaining me.
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Contents
1.
Introduction ........................................................................................................................................
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1.1
Motivation...................................................................................................................................14
1.2
Reviewing various THz generation methods..........................................................................
15
1.3
Considerations for the formulation of an accurate Theory ...............
....................................
16
1.4
Description of thesis...............................................................................................................
17
2.
Physical description of the problem and general approach ..........................................................
19
3.
Optical rectification in the undepleted limit...................................................................................
24
4.
5.
3.1
Effect of tilt angle ........................................................................................................................
3.2
Effect of absorption.....................................................................................................................27
3.3
Effect of group velocity dispersion of the optical pulse................................27
3.4
Effect of optical pump intensity .............................................
29
3.5
Summary .....................................................................................................................................
29
26
Limitations to THz generation by optical rectification......................................................................30
31
4.1
Theory .........................................................................................................................................
4.2
Validation of model: Comparison to experimentally measured optical spectra ............... 32
4.3
Limitation of conversion efficiency due to cascading effects .................................................
34
4.4
A time-domain description ....................................................................................................
37
4.5
Limitations to optical rectification for 800nm pumping ........................................................
38
The 2-D depleted model .....................................................................................................................
5.1
Theory .........................................................................................................................................
40
40
5.1.1
Propagation of optical pump fields through the optical setup.......................40
5.1.2
Nonlinear polarization due to optical rectification ........................................................
5.1.3
Solving the 2-D non-linear wave equation using Fourier decomposition...................... 45
5.1.4
Calculating transmission and propagation of THz radiation .........................................
43
47
5.2
Validation of the model .........................................................................................................
5.3
Understanding effective length in two dimensions................................................................ 49
5.4
Considerations for optimizing conversion efficiency....................................51
48
6.
Conclusion and future work............................................................................................................
53
7.
Appendix A: Tilted-pulse-fronts ....................................................................................................
54
8.
Appendix B: Energy conservation in optical rectification ...............................................................
55
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9.
Appendix C: Num erical convergence of depleted m odel ...............................................................
10. Appendix D: Notes on 2-D im plem entation ...................................................................................
56
58
11. References...........................................................................................................................................61
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List of Figures
Figure 2-1 : (a) An optical pump pulse with phasor E, (c, x0 ,zo) is incident on an arbitrary setup to generate a
tilted-pulse-front. The angular dispersion of various spectral components can generate THz radiation inside the
nonlinear crystal via optical rectification by satisfying the appropriate phase-matching condition. From a timedomain viewpoint, the angularly dispersed pulse forms a tilted-pulse-front as shown by the red ellipse. THz
radiation is generated perpendicular to this tilted-pulse-front. (b) The most common setup used for tilted-pulse-front
optical rectification is a combination of a diffraction grating and lens (c) Time domain snapshots of tilted-pulsefronts in the (x-z) co-ordinate system from Fig. 2-1(a). The intensity-fronts are parallel to the output facet but the
propagation direction is oblique to this. ......................................................................................................................
19
Figure 2-2 (a). Spatial distribution of the optical and THz fluences. The THz field propagates in the z direction with
the optical field at an angle y with respect to it. THz is only generated over a small portion of the optical field due to
spectral broadening of the optical pulse by cascading which results in phase mismatch for subsequent portions of
the beam (b) Optical spectrum is broadened between (i)-(iii) due to cascading effects. (c) THz spectra at locations
(i)-(iii) shows significant spatial-chirp. (d) Since each frequency component has a certain value of transverse
momentum, spectral broadening also necessarily results in broadening in transverse momentum k.....................21
Figure 2-3 : Spatio-temporal break-up of the optical pulses due to simultaneous broadening in frequency and
transverse momentum. Due to spatio-temporal break-up, different parts of the optical pump arrive at different times,
preventing further coherent build-up of THz radiation. .........................................................................................
22
Figure 3-1 : THz spectrum and electric field waveform generated by optical rectification using tilted-pulse-fronts in
the phase-matched condition. The pulse-front-tilt angle is y- 630 at 300 K. THz absorption and group velocity
dispersion of the optical pulse is neglected. Due to small dispersion of the refractive index in the THz range, a broad
band spectrum is generated. The corresponding electric field waveform is an intense 2 ps single-cycle THz pulse
with the peak strength on the order of-M V/cm....................................................................................................
26
Figure 3-2. (a) Effect of tilt angle on conversion efficiency: Small changes in the tilt angle, can lead to large
deviations in the conversion efficiency. This is particularly important while cryogenically cooling the crystal. The
refractive index changes must then be compensated by appropriate changes to the setup. Note that the undepleted
model is strictly not applicable in the above situations since the conversion efficiencies are very large. (b)
Absorption is included into the calculations under phase-matched conditions. The peak efficiencies drop from the
70% level at 300 K to < 10% due to the large absorption of THz radiation at room temperature. After a certain
propagation distance, the amount of generated THz saturates. At 100 K, the effects of absorption are alleviated due
to a reduction in the absorption coefficient. ..........................................................................................................
26
Figure 3-3(a) Effects of group velocity dispersion of the optical pump pulse: In the absence of absorption, the
reduction in peak intensity of the pulse leads to a saturation of the conversion efficiency. The saturation occurs at
shorter distances for larger bandwidths or larger values of group-velocity dispersion. (b) Numerical solution of (3.1),
in the presence of absorption, there is also a decay of conversion efficiency with distance, which is larger for larger
temperatures. This is because as the intensity reduces due to dispersion, absorption begins to dominate, leading to a
reduction of conversion efficiency. .............................................................................................................................
28
Figure 4-1 : Comparison of experimental and simulated optical spectra for different amounts of generated THz. The
frequency down-shift and spectral broadening can be modeled only by the simultaneous solution of the THz and
optical fields. (a) No broadening or red-shift is observed when conversion efficiency j7 = 0 % , which implies small
effect of SPM. (b)There is large cascaded frequency down-shift and spectral broadening corresponding to a larger
amount of THz generation (c) Maximum frequency down-shift and spectral broadening is observed when
conversion efficiency is maximum at 1,.=0.8%. Conversion efficiency, amount of frequency down-shift and
spectral broadening are in good agreement with experiments [14]. The difference between theory and experiment
10
can be attributed to uncertainties in material parameters (Zf,n 2 ,amz) and experiments. The measured spectra
include spatial averaging effects which are not included in the calculations, which could explain their difference... 33
,
Figure 4-2: Conversion efficiencies as a function of effective length are calculated by switching on and switching off
various effects. Material dispersion and absorption are considered for all cases. The pump fluence is 20 mJ/cm2
S = 360pm /V , n2 =10' 5 cm2/W and crystal temperature is lOOK.(a) Gaussian pulses with 500 fs pulse width at
FWHM with peak intensity of 40 GW/cm 2 are used. Cascading effects together with GVD-AD leads to the lowest
conversion efficiencies. The drop in conversion efficiency is attributed to the enhancement of phase mismatch
caused by dispersion due to the large spectral broadening caused by THz generation (See Figs. 4-1(b)-(c)).
However, since group velocity dispersion due to angular dispersion (GVD-AD) is more significant than group
velocity dispersion due to material dispersion at optical frequencies in lithium niobate, cascading effects in
conjunction with GVD-AD is the strongest limitation to THz generation. SPM effects are much less detrimental
since they cause relatively small broadening of the optical pump spectrum (see Fig. 4-1(a)). (b) Cascading effects
along with GVD-AD are most detrimental even for a 150 fs Gaussian pulse with three times larger peak intensity. 35
Figure 4-3: Snapshots of spatial profiles of optical pump pulse at various times. (a)-(c) include only SPM effects and
dispersion (GVD-AD and material dispersion). Dispersion spreads the pulse in time, reducing intensity which leads
to saturation of conversion efficiency. (d)-(f) include only cascading effects and dispersion. The pulse is dispersed
much more rapidly compared to (a)-(c) due to the large spectral broadening........................................................37
Figure 4-4. (a) Experimental curves [29] indicating the relative broadening due to SPM and cascading effects for
pumping by 150fs Ti: Sapphire lasers centered at 800 nm wavelength. Even for these shorter wavelengths/pulse
durations, cascading effects dominate SPM due to the larger extent of broadening. (b) In addition to cascading
effects, three-photon absorption also plays a critical role in impeding THz generation. As the intensity gets higher,
three-photon absorption of the optical pump increases leading to a larger amount of free-carrier absorption of THz
radiation .......................................................................................................................................................................
38
Figure 5-1: 2-D computational space for solving coupled nonlinear wave equations for optical rectification.
Nonlinear crystal geometry is accounted for by delineating an appropriate distribution of e)(xz). Edges of the
distribution are smoothed out to avoid discontinuities. The refractive index is homogeneously distributed throughout
the computational space. The optical beam is centered at a distance h from the apex of the crystal which sets the
limits to the computational region. The optical field at the beginning of the lattice is calculated analytically using
dispersive ray pulse m atrices.......................................................................................................................................43
Figure 5-2(a): Simulation of conversion efficiency as a function of imaging conditions. The surface plot shows the
conversion efficiency versus the imaging distances sl, s2 which are the lens to grating and lens to crystal distances
respectively. The inset shows conversion efficiency versus incidence angle to diffraction grating. (b) Theoretical
calculations of optimal imaging conditions for various pulse-front-tilt angles. For a pulse-front-tilt angle of -63
degrees, the imaging conditions are in close agreement with the simulation results, validating the accuracy of the
presented model. (c) The experimental and theoretically calculated spectra are presented. (d) The output optical
spectrum is presented along with experimental results. ........................................................................................
49
Figure 5-3(a). Effective length in two dimensions. The beam position h and beam position o affect the amount of
absorption and dispersion and affect the effective length in 2-D. Absorption is larger for larger h since there is less
overlap with the optical beam, whereas a is proportional to overlap and therefore reduces absorption. An increase in
h or o increase the effective propagation distance of the optical beam and therefore increase dispersive effects.
Therefore for (b) small h and o, there is minimal absorption (c) for large h and small a, absorption increases and
conversion efficiency drops (d) for large a, only small portions of the beam are involved in THz generation, leading
to an overall drop in conversion efficiency. This has important implications on the scaling of THz energies, by
merely scaling beam size.............................................................................................................................................51
Figure 5-4 (a) Conversion efficiency saturation curves for different beam positions h. The saturation energy/fluence
is dependent on the beam position. (b) While conversion efficiency may increase with increasing intensity, other
11
detrimental effects such as significant spatial-chirp of the THz spectrum also occur. Therefore, optimization of such
systems must consider other aspects of THz radiation while optimizing the setup.................................................52
Figure 9-1. Numerical convergence of simulation for (a) THz spectral phase. (b) THz spectral magnitude.(c) Optical
phase and (d) Optical spectrum for propagation over a distance of 4mm in Lithium Niobate...............................57
Figure 10-1 : Convergence of optical spectra calculations via brute-force (D-1) and (D-4) and Fourier transform
approaches (D-3) and (D-5). The propagation length was 2mm. Beyond this propagation length, the brute-force
approach is numerically too cumbersome to evaluate ............................................................................................
59
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1. Introduction
1.1 Motivation
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To set this thesis into context, it is instructive to describe the ultimate goal at the outset which is the
development of a table-top source of coherent X-ray pulses. The first question then is perhaps why
coherent X-rays? First, short bursts of coherent X-rays can be used to study a number of scientific
phenomena, previously not possible. A large number of reactions in chemistry and biology occur at
extremely short timescales (ultrafast processes). Currently, these reactions are studied with optical
spectroscopic techniques, which obfuscate many of the finer details of the science governing these
processes. A coherent X-ray source would enable study of such systems with unprecedented detail [1]
with atomic spatial resolution and unravel important details about various energy conversion processes.
One example is the study of photosynthesis, a deeper understanding of which will have important
implications on the long term solution to the societal energy problem.
In addition, coherent X-rays would open up the possibility of clinical phase-contrast imaging using Xrays. Current X-ray imaging modalities employ incoherent X-rays and use their attenuation as the contrast
mechanism. This limits visualization of features with similar attenuation (e.g. tumors and surrounding
soft tissue). With coherent X-rays, the phase alteration imparted by tissue can be measured instead, since
phase differences in different soft tissues are -1000X greater than their attenuation differences. Initial
experiments show that phase contrast can for the first time, distinguish normal from cancerous tissue,
characterize components of atherosclerotic plaque (the main culprit behind heart attacks and strokes) and
show soft tissue anatomy with detail unrivalled by any existing imaging modality. Accurate diagnosis is
therefore expected to have a huge impact since cardiovascular disease and strokes alone incur -300
billion US dollars [2] while cancer incurs 228 billion US dollars in the US annually. Radiation therapy
using hard X-rays is another important problem that can benefit from a coherent X-ray source. Current
therapy using incoherent X-rays are limited in their ability to localize radiation to the region to be treated
which leads to unwanted side effects due to radiation dose to neighboring tissues. With a coherent source
of hard X-rays, it would be possible to generate a highly accurate radiation therapy plan for highly
localized treatment.
Next we ask, how does one achieve this target? As always, multiple approaches exist. Higher harmonic
generation (HHG) is currently the most widely pursued approach but has low conversion efficiencies
between 10-9 and 1012. It therefore makes sense to look for alternative approaches, since existing
approaches have not yielded significant returns over several years. An alternative approach, with potential
to achieve conversion efficiencies of 10-4, is to create a bunch of coherently accelerated electrons, with
many Giga electron-volt of energy (GeV) and then decelerate them coherently, all at the same time to
create a source of coherent X-rays. The acceleration would have to be especially compact for this
approach to be feasible. Typically, accelerating electrons up to a 10 GeV using current radio frequency
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based linear accelerators (LINACS) would require a distance in the kilometer (km) range. If, the size of
such structures can be shrunk by a factor of 100, to -10m, it would be possible to house them in
laboratories or clinics. Such compact electron sources in themselves are extremely useful for a number of
applications - once again in the areas of medicine (skin therapy [3] ), imaging and scientific investigation
[4].Therefore in addition to the goal of generating coherent X-rays, there are other short term rewards to
be reaped.
The current RF sources with wavelengths of a few tens of cm have low peak electric fields, which
prevents them from imparting a strong force to the electron. Consequently, a long distance on the order of
-kilometers (km) is necessary for accelerating electrons upto a several GeV of energy. Using intense
electric fields of optical pulses is one way to accelerate electrons over very short distances. However,
since the optical wavelength is very short (- pm), the slow -moving electron lags the electric field of the
fast propagating light and is likely to experience an equally powerful retardation due to the negative halfcycle of the optical electric field. Therefore, while the force the electron experiences is very large, the
acceleration is not very efficient because of the short wavelength of the optical field.
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Wavelengths, approximately 100 times smaller than RF and a hundred times larger than optical radiation,
could serve as a sweet spot for accelerating electrons by leveraging on the relative advantages of both
kinds of radiation- long wavelengths similar to RF radiation and large electric fields similar to optical
radiation. Theoretical calculations [5] reveal that acceleration of electrons from rest to 10 MeV energy
levels would require radiation at -500 pm with 10 mJ pulse energy and pulse duration tunable in the
range of a few ps (for acceleration from rest to the MeV or relativistic level) to a few hundred ps (for
acceleration from the MeV to GeV level). This would result in Giga-Watt (GW) peak powers and peak
electric fields in the GV/m range or higher. This region of the electromagnetic spectrum with wavelengths
between 3 mm (100 GHz) and 3 pm (10,000 GHz) is generally defined to be the terahertz (THz) region of
the electromagnetic spectrum. Once again, a source of THz radiation with 10 mJ pulse energy and a GW
of power in itself has many rich applications. The most relevant area is that of nonlinear spectroscopy
i.e, the investigation of the properties of a number of different molecular and solid-state materials [6].
1.2 Reviewing various THz generation methods
Initially, considered a 'forbidden zone' due to its location at the extremities where both electronic and
optical technologies reach their limits, a number of continuous wave (CW) and pulsed THz sources have
been developed to fill the so called 'THz gap'. It is worthwhile to examine these various technologies in
order to ascertain which approach would enable us to get to the goal of 10 mJ pulse energy with pulse
durations in the few to hundred picosecond range and intense electric fields.
One broad category of CW sources comprises of solid-state devices which up-convert frequency [7]- [8].
Such sources produce radiation in the lower range of THz frequencies (0.1 to 1 THz). Quantum Cascade
Lasers (QCL's) form another important category of CW THz sources, predominantly for higher THz
frequencies (1-10 THz) [9]. Solid-state photomixers are the most commercially ubiquitous technology
and can produce broadband THz pulses (- ps duration) as well as CW radiation in the (0.1 to 3 THz)
range [7]. However, the above sources are limited to low THz peak fields and cannot be scaled to the
desired 10 mJ level pulse energies.
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Other sources of THz radiation are vacuum electronic devices (VED's) [10] which can produce peak
output powers anywhere from a kW to a GW (free-electron-lasers). However, output from VED's tend to
be at the lower THz frequency range of - 0.1 THz. While, travelling wave tubes can reach higher THz
frequencies, they are limited in bandwidth (a few nanoseconds of pulse duration) and peak powers (MW).
Free-electron lasers produce radiation with the desired properties but are expensive, inaccessible and not
amenable for deployment in conventional research laboratories.
Down-conversion of optical frequencies using nonlinear optical crystals has emerged as a potential
approach to generating THz pulses with peak powers, bandwidths and fields in the desired range. The
feasibility of the approach is enabled by the development of relatively cost-effective and robust table-top
solid-state laser technology, producing upto 100 mJ pulse energies at kHz repetition rates in the optical/
infrared (IR) spectral region.
A number of crystals - gallium arsenide, gallium phosphide, zinc telluride, cadmium sulphide, lithium
niobate and organic crystals - DAST/DSTMS [11] are potential candidates for crystals in such non-linear
down conversion processes. However, nonlinear absorption processes in various crystals impose specific
requirements of the optical pump wavelength. Of various solid-state laser technologies, lasers centered at
800 nm or 1 pm are the most mature and therefore it makes sense to exploit these advancements for THz
generation. An important material system amenable to pumping by these wavelengths is lithium niobate.
It also possesses amongst the highest second order nonlinear susceptibility values.
Of various nonlinear optics based approaches, optical rectification (OR) or intra-pulse difference
frequency generation (DFG) using tilted-pulse-fronts in lithium niobate has emerged as the most efficient
approach [12]- [13]. Optical-to-THz conversion efficiencies in excess of 1% [14], THz pulse energies of
0.4mJ [15] have been demonstrated. These promising initial results have attracted significant interest in
further scaling of the approach. In addition, theoretical calculations predict further improvements to
conversion efficiency and pulse energies [16].
1.3 Considerations for the formulation of an accurate Theory
Theoretically, up to 10% conversion efficiency has been predicted for the above approach [16]. This
translates to THz pulses with pulse energies of 10 mJ or peak powers of several GW. However,
experimental demonstrations have only produced conversion efficiencies of a few percent at best, despite
the use of theoretically prescribed conditions. The significant gap between theory and experiment poses
important questions about why the demonstrated efficiencies are much lower than expected and suggests
a possible gap in the current level of understanding of the approach. Therefore, a re-examination of
existing theory is warranted.
A few key physical effects have to be included for satisfactory modeling of THz generation via optical
rectification using tilted-pulse-fronts. These effects have been omitted in the present models in the
literature.
(i)
In optical rectification using tilted-pulse-fronts, an optical pump pulse is angularly dispersed
to produce an intensity front which is tilted with respect to its propagation direction. The
generated THz radiation propagates perpendicular to this tilted intensity front or tilted-pulse16
(ii)
(iii)
(iv)
front. Since the optical and THz radiation travel different distances in the same time, the large
difference between optical and THz refractive indices is compensated and phase matching is
achieved. As a consequence of the angular dispersion of the optical pump pulse, various
frequency components of the optical pump pulse spectrum are now spatially separated and
have different amplitudes and phases at each spatial location. This is tantamount to having
different spectral bandwidths and pulse durations at each spatial location. These spatiotemporal distortions of the optical pump pulse are expected to affect the properties of the
generated THz radiation and have to be accounted for.
Since the generated THz propagates perpendicular to the tilted-pulse-front, the optical pump
and THz radiation propagate non-collinearly in different directions. Therefore, the problem
can only be satisfactorily described in a minimum of two spatial dimensions (2-D).
As THz radiation is generated, it is accompanied by a dramatic cascaded frequency downshift and spectral broadening of the optical pump pulse spectrum (cascadingeffects) which
has been experimentally modelled but not satisfactorily modelled in theory. While the optical
pump energy is relatively undepleted due to small conversion efficiencies, the spectral reshaping is drastic and therefore expected to play a role.
At very high intensities, in addition to cascading effects, self-phase-modulation maybe an
important factor and would have to be modelled.
A comprehensive theoretical model should therefore be able to account for all of the above effects. This
would require a simultaneous solution of optical and THz electric fields (henceforth referred to as field) in
at least two spatial dimensions (2-D). In addition, spatio-temporal distortions imparted by the imaging
optics should also be considered.
Previously presented models broadly comprise of (i) 1-D and 2-D spatial models without the inclusion of
cascading effects and (ii) 1-D spatial models which account for cascading effects. Of works in the first
category, a one-dimensional (1-D) spatial model including the effects of material dispersion and group
velocity dispersion due to angular dispersion (GVD-AD) was presented [17]. It included a Drude model
for free-carrier-absorption (FCA) of THz radiation to account for the saturation of THz energy. In [18], a
I-D model considering material dispersion, GVD-AD and self-phase modulation (SPM) was presented. In
[19], a 2-D model which took into account material dispersion, GVD-AD and crystal geometry was
developed. In the second category, an effective 1-D model with cascading effects was first presented in
[20] and improved in [21]- [22].None of the presented models include the effects of spatio-temporal
distortion of the optical pump pulses or cascading effects in 2-D.
1.4 Description of thesis
In Chapter 2 of this thesis, we motivate the need for the development of a comprehensive theory, by
illustrating the physics at work in optical rectification using tilted-pulse-fronts.
Chapter 3, initiates the reader to THz generation by optical rectification using tilted-pulse-fronts by
simple models with undepleted pump approximations. Simple analytic expressions are derived which
bring out certain essential features of the model. These include the nature of the generated THz spectrum,
the effects of dispersion and absorption. The chapter ends with a discussion on its failure to account for
the saturation of conversion efficiency, which is experimentally observed.
17
In Chapter 4, we solve simultaneously for both optical and THz fields. We show that the experimentally
observed cascaded frequency down-shift and spectral broadening is a direct consequence of THz
generation. It is shown that this spectral broadening accentuates dispersive effects and ultimately limits
further THz generation. This is an important finding in that it indicates that optical rectification using
titled-pulse-fronts is a self-limiting process. The contents of this chapter also constitute a journal
publication.
In Chapter 5 of this thesis, we present a detailed formulation of the model which simultaneously accounts
for spatio-temporal distortions of the optical pump pulse, cascading effects, SPM and Stimulated Raman
Scattering (SRS) effects, material dispersion and absorption as well as nonlinear crystal geometry in 2-D.
We validate the model by comparisons to theory and experiment and provide guidelines on how to
optimize the experiment for conversion efficiency. We find that the increase in conversion efficiency is
not always desirable as other properties such as beam quality are also affected. This lays the foundation
for more careful optimization of THz sources based on optical rectification.
We conclude in Chapter 6, laying the roadmap for future work.
18
2. Physical description of the problem and general approach
In this chapter, we provide an overview of the physics at work in optical rectification using tilted-pulsefronts. This should serve to motivate the theoretical development of the remainder of the thesis. The
overall schematic of our approach is depicted in Fig. 2-1(a). An optical pump pulse with input electric
field described by E'(w,xo,zo)at angular frequency co, propagating in the zo direction is incident on an
arbitrary setup for creating tilted-pulse-fronts. The co-ordinate axes x0 and zo are as depicted in Fig. 21(a). In Fig. 2-1(a), electric fields at only two angular frequencies -co and w+Q are depicted for
convenience although there are an infinitely large number of frequency components in the optical pump
pulse spectrum. The tilted-pulse-front setup can be chosen arbitrarily could include a grating and lens, a
grating and a telescope, a contact grating etc. The most commonly employed setup for generating tiltedpulse-fronts is comprised of a diffraction grating and a single lens and is depicted in Fig. 2-1(b).
Optical Beam
)
E" (co+f, x, zo)
E n (co+Q,x0 , z0
Tilted Pulse Front
setup
E
E /(w,x 0 ,zo)
ilted Pulse Front
~()
i(O+X Zd 0
T z
(+x 0 0)
'THz Output Field
(ETz(oX,Zd)
I
(a)
Diffraction
Gratingpt
QpfIcal Beam
Tenpoal Snapshotw of optical Tilted Pulse Fronts
Tilted Pulse Front
k (o)
-
k(.o)
n/2-Y
(b)
y
V
L
A
" xW in)
'
THz
(c)
Figure 2-1 : (a) An optical pump pulse with phasor E" (w, x0 , z.) is incident on an arbitrary setup to generate a tiltedpulse-front. The angular dispersion of various spectral components can generate THz radiation inside the nonlinear
crystal via optical rectification by satisfying the appropriate phase-matching condition. From a time-domain
viewpoint, the angularly dispersed pulse forms a tilted-pulse-front as shown by the red ellipse. THz radiation is
generated perpendicular to this tilted-pulse-front. (b) The most common setup used for tilted-pulse-front optical
rectification is a combination of a diffraction grating and lens (c) Time domain snapshots of tilted-pulse-fronts in the
19
(x-z) co-ordinate system from Fig. 2-1(a). The intensity-fronts are parallel to the output facet but the propagation
direction is oblique to this.
The optical pump pulse is incident at an angle 0, with respect to the normal of the grating surface and
each frequency component of the optical pump pulse with angular frequency c) scatters off at a different
angle 6 d(w). A focusing lens, located at a distance s, from the diffracting grating is used to image this
angularly dispersed optical pump pulse onto the lithium niobate crystal located at a distance s 2 from the
lens as shown in Fig. 2-1(b). An optical pulse with a tilted-pulse-front is typically angularly dispersed
[23]. Therefore, various frequency components of the emergent optical field described by the phasor
EO'(),xO,zo) propagate at different angles. This is depicted in Fig. 2-1(a)-(b) as the spectral components
at w and c)+Q emerge with corresponding wave vectors k(a),k(a+0) respectively.
Since optical
rectification is essentially difference frequency generation (DFG), in the phase matched condition, the
propagation vector of the generated THz (at angular frequency
) is given by k(l) = k(w + Q) - k(a) as
seen in Fig. 2-1(a)-(b). The generated THz then emerges at an angle y with respect to the direction of
propagation of the optical pump pulse and exits approximately normal to the output facet of the crystal.
In the time domain, such an angularly dispersed pulse has an intensity profile which is tilted with respect
to its propagation direction as delineated by the red ellipse in Fig. 2-1(a) which gives rise to the
terminology of 'tilted-pulse-front'. This red ellipse makes an angle r/2-y with respect to the propagation
direction of the optical pulse, where y is referred to as the pulse-front-tilt angle.
This is depicted via
temporal snapshots of the intensity profile of an angularly dispersed optical pulse in Fig. 2-1(c), where the
pulse propagates obliquely but has an intensity profile parallel to the output facet. A theoretical
description of tilted-pulse-fronts and their relationship to angular dispersion is provided in Appendix A.
Since, the various frequency components are angularly separated, they have separating field profiles
which leads to a spatial variation in the average frequency as well as relative amplitude and phase across
the beam profile. As a consequence, the pulse durations and bandwidths at various points in space are
different, which in turn has implications on the spatial properties of the generated THz pulse. These
spatial variations of frequency and pulse duration as a consequence of angular dispersion and spatial chirp
are categorized as spatio-temporal variations. In Chapter 5, we consider these spatio-temporal effects by
applying an analytic formulation of dispersive ray pulse matrices [24]. This approach efficiently models
the various spatio-temporal effects associated with the optical pulse for an arbitrary tilted-pulse-front
setup and supplies Eg'(w,xO,zo)at the input face of the nonlinear crystal. In chapter 5, this is then used as
an initial condition for the solution of a system of 2-D coupled nonlinear wave equations in the nonlinear
crystal to obtain both THz and optical field profiles.
20
(a)
Optical Spectra
0.5-
(b)
I
(i)
40.5
-4
I
I
0
(ii)
-
Z 0.25
-0.5
(iii)
0.25
1
0
2
1000
1020
z(mm)
THz Spectra
0.2
1040
1060
1100
(d)
(c)
k Distribution of Optical Pump Fluence
(i)
z
0.1
=
-
0.3mm
0.5-
0
1
0
0.2-
z = 0.54mm
-
0.5
0.1-
C0
0
0.5
1080
Wavelength (nm)
.1
(iii
I
z = 1.4mm
0.05I
0
0.5
1
1. 5
-1.5
k(x 10)5.
0
0.5
Frequency (THz)
Figure 2-2 (a). Spatial distribution of the optical and THz fluences. The THz field propagates in the z direction with
the optical field at an angle y with respect to it. THz is only generated over a small portion of the optical field due to
spectral broadening of the optical pulse by cascading which results in phase mismatch for subsequent portions of
the beam (b) Optical spectrum is broadened between (i)-(iii) due to cascading effects. (c) THz spectra at locations
(i)-(iii) shows significant spatial-chirp. (d) Since each frequency component has a certain value of transverse
momentum, spectral broadening also necessarily results in broadening in transverse momentum k,.
In Fig. 2-2, we provide a sample solution of the system of 2-D nonlinear equations using the developed
model. Fig. 2-2(a), depicts the optical and THz fluences obtained. Regions within the green lines contain
nonlinear material with a non-zero second order susceptibility. In the (x-z) co-ordinate system, the optical
pump pulse propagates obliquely in the nonlinear crystal as indicated by the cyan/light-blue regions in
Fig. 2-2(a). The THz pulse however propagates in the z direction, at an angle y to the optical pump pulse
and emerges approximately perpendicular to the output face as seen by the THz fluence in red.
21
As the optical pulse propagates, it generates a THz photon at
and simultaneously suffers a frequency
downshift by the same amount. Upon each successive generation of a THz photon at , it repeatedly
experiences a frequency down-shift or a 'cascaded' frequency down-shift which leads to a large spectral
broadening of the optical pump pulse spectrum as seen in Fig. 2-2(b). However, as the optical pump pulse
spectrum is modified, the subsequent THz spectrum is also modified as shown in Fig. 2-2(c) and viceversa. As a result, there is a great spatial variation in both optical and THz spectra as seen in Fig. 2-2(b)
and 2-2(c). The spatial-chirp of the THz spectrum has important implications on beam transport and
focusing properties. This spatial-chirp in THz and optical spectra has been observed experimentally [25]
but not explained theoretically, until now.
For an angularly dispersed pulse, each spectral component of the optical pulse at co has a well-defined
transverse momentum k. Therefore, spectral broadening of the optical pulse also leads to a re-distribution
of the optical pulse energy among various transverse momentum (k,) values. Therefore, as the optical
pulse propagates, it also broadens in transverse momentum. Eventually, as the optical spectrum is
broadened as seen in Fig. 2-2(b), the angular dispersion is accentuated to such a great extent that phase
matching is lost and THz generation ceases altogether. This is seen in Fig. 2-2(a) where THz is generated
only over a small distance of optical pump pulse propagation. Previous theoretical treatments have
ignored this spectral broadening of the optical pump pulse in their calculations which has led to a gross
overestimation of the achievable conversion efficiency [16]. This finding has very important implications
on the scalability of this approach. This large broadening of the optical pump pulse spectrum causes a
temporal break-up of the pulse. Similarly, due to broadening in transverse momentum, there is also a
spatial break-up of pulses. In combination, a rapid spatio-temporal break-up of the optical pulse occurs as
shown in Fig. 2-3. Thus, different parts of the optical pulse arrive at different time instants and the
generated THz no longer builds up coherently. This physical picture, based on the spatio-temporal breakup of pulses due to broadening in transverse momentum and frequency, is a new lens to look at the THz
generation process.
0.25
2
0.2
0
0.15
E
E -2
0.1
-4
0.05
-6
-1
0
1
2
-1
0
1
2
-1
0
1
2
0
z (mm)
Figure 2-3 Spatio-temporal break-up of the optical pulses due to simultaneous broadening in frequency and
transverse momentum. Due to spatio-temporal break-up, different parts of the optical pump arrive at different times,
preventing further coherent build-up of THz radiation.
22
Thus, in this thesis, a few important findings are presented. First, it resolves the question as to why THz
conversion efficiencies are not as high as predicted. Next, it provides a detailed physical description of the
optical rectification process and identifies the rapid spatio-temporal break-up of optical pulses as the key
limiting factor to THz generation. This new physical picture has important implications on the scaling of
optical pump energies and beam sizes. It also informs us on what factors need to be considered while
optimizing the setup.
23
3. Optical rectification in the undepleted limit
In Chapter 2, we provided an overview of the physics at work in the process of THz generation by optical
rectification using tilted-pulse-fronts. In order to build up to the level of understanding required to
accurately model such systems, it would be useful to start with models where the undepleted pump
approximation is invoked. Since the experimentally observed optical to THz conversion efficiencies (or
conversion efficiency for short) are < 10%, this would seem to be a reasonable approximation at first. It
will be seen in this section that such models are in fact sufficient to explain many essential features of
THz generation by optical rectification, at least qualitatively.
Although THz generation using tilted-pulse-fronts is a non-collinear process and can be modeled
satisfactorily only in two dimensions (2-D), we begin with an analysis using an effective 1-D equation,
similar to [26]. Let the electric field of THz radiation with angular frequency Q be z propagating and
denoted by Em,(n,z). The optical pump pulse propagates at an angle y with respect to the THz pulse,
where y is the pulse-front-tilt angle. In order to reduce the problem to a single dimension, it is useful to
note that if the THz field propagates a distance z, the optical field propagates a distance z/cosy or
alternatively, the effective wave number of the optical field may be divided by a factor of cosy. Such a
model would be phenomenological, but it is effective in providing an understanding of many of the
features of optical rectification using tilted-pulse-fronts.
Keeping in mind the above, we write the 1-D wave equation for the THz electric field at each angular
frequency 2 driven by the nonlinear polarization term PTH (f,
d 2E
(,,z) +2
dz
z)
2Em,(~z)(3.1)
+k(G)E,(n~)=
2
60 C
The nonlinear polarization term for THz radiation at an angular frequency for optical rectification is
basically an ensemble of all possible difference frequency generation processes within the bandwidth of
the optical pump pulse spectrum. Consequently, it may be written down according to (3.2), where Z4 is
the effective second-order nonlinearity of the material.
P(C2, z) = 60
zf
A, (co + n)A, *(c)e-JIkC"*n)~'o))zd()
(3.2)
In (3.2), A.,(w) is the envelope of the electric field of the optical pump pulse at angular frequency co. The
factor in the exponent k(w + Q) - k(w) = (cos y)-l n, (co) / c when the effects of group velocity dispersion
are excluded. It phenomenologically represents the tilted-pulse-front. For a transform-limited Gaussian
pulse centered at co, the expression in (3.2) reduces to that in (3.3).
nI22
E2
(3.3a)
PT.( )=z or2
e
T
2,_,-,o
A,( ) ,c)
= Er e
0 0 2 r eY
4
2
(3.3b)
24
In (3.3), E is the peak electric field amplitude and r / V[ is the e- pulsewidth. It can be related to the
full width at half maximum (FWHM) pulse width by the relation rFwHM / V2n 2 = r . A general analytic
solution to (3.1) is possible in the limit that group velocity dispersion (GVD) of the optical pulse is
neglected. It is instructive to examine this expression as it provides significant insight into the properties
of THz radiation generated via optical rectification. In general, the solution to (3.1) is a sum of a
homogeneous solution and a forced solution. This is expressed below in (3.4). The backward propagating
THz wave is assumed to be zero. This is a reasonable approximation in sufficiently thick crystals where
the backward propagating THz wave would be absorbed by the material.
2
ET.Hz (rl,z) = Gek(a)z
EmzQ~z=Ce72C)"
ffEO e _~e
e
Cb,
_Z
(34
(3.4
c222i
The constant C, is resolved by applying the boundary condition
Emz
(0, 0)= 0 which signifies that the THz
field is zero at the beginning of the crystal. Including these arguments, we have the following analytic
expression for the THz field in (3.5).
n2,2
E22(, z)
=
-r
J.1rb(a)+rb,(ft)
sinc
+
c22;r
-z[n(9)-lg(w))/cosylzJ
(3.5)
nC
)
C cosy
Equation (3.5) is similar to those from various other nonlinear optical generation processes [27]. Its
variation with propagation is characterized by the typical sinc function, which is maximum for an
argument of zero. This condition dictates the phase-matching condition. Therefore, setting the argument
of the sinc function in (3.5) to be zero, we obtain the corresponding phase-matching condition in (3.6).
n(C2)cosy = ng(wo)
(3.6)
This means that the pulse-front-tilt angle y must be adjusted such that (3.6) is satisfied. In lithium niobate,
where the THz refractive index is -5 and the group refractive index at optical wavelength is -2.25, the
phase-matching angle is -63 degrees. If this phase-matching condition is satisfied, the THz electric field
sees a linear growth with distance. Adopting a time-domain viewpoint, what this means is that as the THz
radiation is generated, the optical field propagates at an angle y to it and the subsequently generated THz
radiation adds up coherently with the previously generated THz radiation. In lithium niobate, the
dispersion of refractive index in the lower THz range is relatively low and consequently a very broadband THz pulse is generated. This is the first salient feature of THz generation by optical rectification
using tilted-pulse-fronts. We plot the generated THz spectrum and temporal waveform below in Fig 3-1.
for a pulsewidth vwm
= 750 fs and fluence of 20 mJ/cm2 , propagated over 2 mm. The pulse-front-tilt
angle y was set to ~630.
25
z 10'Termhertz Spectrum and temporal waveform
x 10 THz Ilectric eld w Vefon
3
5
o
2.5
2-
WL.5
-S
-19
1
0.5
time(ps)
=
tFWM 750f
F
=20
mJ/cm2
,
0pm
-15
-17 -16
-18
0.5
1
1.5
2
Frequency (THz)
2.5
Figure 3-1 : THz spectrum and electric field waveform generated by optical rectification using tilted-pulse-fronts in
the phase-matched condition. The pulse-front-tilt angle is y- 630 at 300 K. THz absorption and group velocity
dispersion of the optical pulse is neglected. Due to small dispersion of the refractive index in the THz range, a broad
band spectrum is generated. The corresponding electric field waveform is an intense 2 ps single-cycle THz pulse
with the peak strength on the order of-MV/cm.
3.1
Effect of tilt angle
One of the key features of tilted-pulse-front phase matching is the high sensitivity of the generation
process to the tilt angle of the pulse-front. This sensitivity makes the alignment of the setup critical to
both conversion efficiency and the spectral properties of the THz radiation. In the actual experimental
setup, y is controlled by the grating angle and the magnification of the imaging system. Here, we
artificially vary it to observe the effects on conversion efficiency as shown in Fig.3-2 (a). The conversion
efficiency can be calculated according to (3.7).
;rce fn(Q) I E (n, z)1 d2
(3.7)
0(z)
Fo
=SOfs
=20 mJ/cm
tr
70-F
70
100K
PM =
L = 3mm
t
/40
spton= 0
60
of Absorption
soEffect
Conversion Efficiency Vs Tilt Angle
=7SOfs
20 mJ/cm 2
PWMM2
=
F
30
so-U
100
940
30-
300K
020-
W
20
300K
0 10
10
0
61
Tilt
63
gle (degrees)
64
0
2
6 (
4
Propagation Distance (mm)
10
Figure 3-2. (a) Effect of tilt angle on conversion efficiency: Small changes in the tilt angle, can lead to large
deviations in the conversion efficiency. This is particularly important while cryogenically cooling the crystal. The
refractive index changes must then be compensated by appropriate changes to the setup. Note that the undepleted
model is strictly not applicable in the above situations since the conversion efficiencies are very large. (b)
Absorption is included into the calculations under phase-matched conditions. The peak efficiencies drop from the
70% level at 300 K to < 10% due to the large absorption of THz radiation at room temperature. After a certain
propagation distance, the amount of generated THz saturates. At 100 K, the effects of absorption are alleviated due
to a reduction in the absorption coefficient.
26
3.2 Effect of absorption
If we assume that phase-matching is satisfied in (3.5) but now include an absorption term a,, , then (3.5)
reduces to (3.8a). In the presence of some phase mismatch Ak, the general expression for the field is given
by (3.8b) and the corresponding magnitude is given by (3.8c).
rp2
2
ET2) =
ETHZ(Q~z
-ik
-TH
z/
(3.8a)
2
2~
~Ere
n2,2
2E
r,(,Z)2 =
1-e -an,,L/2
c 2.4 r2~
c24ii2k())
ajMz/ 2 2 +4sin2 2
fe2
(2)Eore
C 2 452k(O)
2
nz
aTk(n)
e1 "e -~
(Ak2 +aTz
-
1T,/2)ekf
+
2
~
(3 .8b)
)
e-
z)=
Er,,(C2,
,12
q
2
(3.8c)
Ak2 +a
.2
4)
This is also quite akin to what is observed in various nonlinear optical generation processes, i.e. there is
an initial increase in the THz field and subsequent saturation. We see this in Fig.3-2(b) for the 300 K case,
when the conversion efficiency saturates at about 8% at z-10 mm. In the phase mismatched case, there is
decay in the field over longer distances instead of mere saturation. Intuitively, one can understand this as
follows. If the THz generation process is not phase matched, there is a sinusoidal cycling of energy
between the optical and THz radiation or on an average, there is no increase in the amount of generated
THz radiation. However, if there is no average increase in the amount of generated THz but there is a
simultaneous absorption of radiation, there is a net decay of THz energy. This is seen in (3.8c) by the
presence of the sin 2 (Akz / 2)e
2
term in (3.8c). A solution to mitigate the effects of absorption is to
cryogenically cool the lithium niobate crystal to 100 K. This reduces the absorption coefficient and in
principle, increases the distance over which THz radiation can be generated as seen in Fig. 3-2(b).
However, care must also be taken to realign the setup to introduce the correct pulse-front-tilt angle, since
the refractive index of the crystal is sensitive to temperature as seen in Fig. 3-2(a).
3.3
Effect of group velocity dispersion of the optical pulse
In the expression in (3.5), we did not consider the effects of group-velocity dispersion of the optical pulse
on the THz generation process. Under this assumption, there is no distortion to either the optical pulse
spectrum or the temporal profile as it generates THz radiation. However, since the pulse has a tilted-front,
it suffers group velocity dispersion due to both angular and material dispersion. Incidentally, for lithium
niobate, angular dispersion is significantly more important by a factor of more than 15. In the presence of
such dispersion, the peak intensity of the pulse drops as the pulse broadens in time during its passage
through the crystal. The drop in intensity is accompanied by a drop in THz generation. The generated
THz is in turn absorbed, leading to decay in conversion efficiency. In addition, due to dispersion, there is
a walk-off between the generated THz and optical pump pulses, which leads to a reduction in the coherent
build-up of THz radiation (even in the absence of absorption). In the absence of absorption, an analytic
expression for the generated THz electric field can be derived for a given group velocity dispersion value
27
6 " in (3.9). An analytic expression for group velocity dispersion due to angular dispersion can be
derived in (3.9b). The derivation of the expression is presented in Appendix A.
ffg E(
2
ETHz, Z) =
2
8
erfl
lg2 (o)tan
=2r
(3.9a)
z
(3.9b)
cocn,(coo)
The error function in (3.9a) erfix), tends to 1 as x tends to infinity. This means that there will be a
saturation of the conversion efficiency for large z. Furthermore, the larger the /3", the smaller the value of
z at which this saturation occurs is shorter. Similarly, the larger the bandwidth (or shorter the value of r)
the earlier saturation of conversion efficiency sets in. We see these effects in Fig. 3-3(a).
Effects of Group Velocity Dispersion of pump
tFWHM=500fs
No Absorption
25 2-
20
=750fs
SFMHM
15t
C10-
=250fs
FWHM =15Ofs
0
0
5
20
15
10
25
z(mm)
THz Absorption and Pump Group-Velocity Dispersion
30C
25-
100 K
------ 300 K
-
20
10-
00
5
10
is
z(mm)
20
25
Figure 3-3(a) Effects of group velocity dispersion of the optical pump pulse: In the absence of absorption, the
reduction in peak intensity of the pulse leads to a saturation of the conversion efficiency. The saturation occurs at
shorter distances for larger bandwidths or larger values of group-velocity dispersion. (b) Numerical solution of (3.1),
in the presence of absorption, there is also a decay of conversion efficiency with distance, which is larger for larger
temperatures. This is because as the intensity reduces due to dispersion, absorption begins to dominate, leading to a
reduction of conversion efficiency.
In Fig. 3-3(b), a numerical solution of (3.1) is presented, taking into account the complete dispersion of
refractive index at both optical and THz refractive frequencies as well as angular dispersion. It is seen that
inclusion of absorption leads to a drop in conversion efficiencies to the sub -10% level at 300 K.
28
Furthermore, there is a decay of conversion efficiency with propagation as no further THz is generated
due to the drop in intensity of the optical pulse. Consequently, absorption begins to dominate.
3.4 Effect of optical pump intensity
From (3.1) to (3.9), it can be seen that there is no provision in the presented theory describing the
limitations to the optical pump intensity. This does not make physical sense, since an arbitrarily high
intensity can then be used to convert all the optical energy to THz energy in a short propagation length
(where absorption has a minimal impact). In practice, self-phase modulation and associated self-focusing
effects as well as optically induced damage present limitations. The limitations to THz conversion
efficiency will be studied in detail in the subsequent chapter.
3.5 Summary
An analytic approach was used to illustrate certain key effects associated with THz generation by optical
rectification using tilted-pulse-fronts. The phase-matching condition was established and the effects of
dispersion and absorption were discussed. However, the model does not account for the saturation of THz
generation with increasing optical intensity. Based on the calculations presented above, it would seem
that conversion efficiencies in excess of 10% are easily achievable. However, in practice the maximum
conversion efficiencies that have been obtained with such systems are an order of magnitude lower
(closer to 1%). Furthermore, experimentally a drastic spectral re-shaping of the optical pump pulse is
seen which may have implications on the properties of the system. These shortcomings of the model are
investigated in the subsequent chapters in greater detail.
29
4. Limitations to THz generation by optical rectification
In Chapter 3, we used undepleted models to illustrate certain important characteristics of THz radiation
generated via optical rectification using tilted-pulse-fronts. A key problem with that theory was that it
does not account for the saturation of THz energy. Therefore, conversion efficiencies of 10% or more at
lOOK are predicted for transform limited 500 fs full-width-at-half-maximum (FWHM) pulses centered at
a wavelength of - 1 pm using such models [16]. On the other hand, experimental results [14], [28] have
not yielded similar values. The disparity between experimental and theoretical results motivates a reexamination of existing models to identify potential shortcomings in theory.
In previous models [17]- [19], the spectral re-shaping of the optical pump pulse caused by THz generation
was not considered. However, these effects have to be included to explain the dramatic experimentally
observed frequency down-shift and spectral broadening of the optical pump pulse, commensurate with the
amount of THz generated, using 1pm [14] as well as 800 nm pump sources [25], [29]. This spectral
broadening can intuitively be understood to be a consequence of a cascaded down-shift of the optical
pump frequency upon generating THz photons. We will henceforth refer to the cascaded down-shift and
spectral broadening of the optical pump pulse spectrum as 'cascading effects'.
In this chapter, we present a 1-D spatial model which does not make an undepleted pump approximation.
The model simultaneously accounts for (i) cascading effects, (ii) material and angular dispersion, (iii)
absorption, (iv) SPM and (v) stimulated Raman scattering (SRS). The presented model is applicable to
simulating a wide variety of OR devices based on different material systems and crystal geometries.
Using the 1-D model, we are able to confirm that the experimentally observed cascading effects are a
direct consequence of THz generation. The extent of spectral broadening is demonstrated to be
commensurate to the amount of THz generated. Furthermore, in both experiments and theory, it is seen
that SPM effects cause negligible spectral broadening. The quantitative agreement of simulated optical
spectra and conversion efficiency with experiments [14] is a good validation of the model.
Due to the large spectral broadening resulting from THz generation, the phase mismatch due to
dispersive effects is significantly accentuated. Consequently, the conversion efficiency decays much more
drastically with propagation length than it would due to other limiting effects such as SPM. Since, GVDAD (-15000 fs 2/mm) is much larger in magnitude than the group velocity dispersion due to material
dispersion at optical frequencies (<1000 fs 2/mm) in lithium niobate, it is the dominant dispersive effect.
Therefore, cascading effects in conjunction with GVD-AD represents the strongest limitation to achieving
high conversion efficiencies in lithium niobate, especially for pumping at 1 sim (or longer wavelengths).
For pumping at 800 nm, in addition to the limitations imposed by cascading effects, it will be shown that
FCA is also an impediment to THz generation [29].
We show that neglecting these cascading effects, i.e. the spectral re-shaping of the optical pump
spectrum, can lead to a significant overestimation of achievable conversion efficiencies. For example,
when cascading effects are excluded, conversion efficiencies as high as 10% have been predicted [16].
However, for similar conditions, the consideration of cascading effects results in conversion efficiencies
of only -2%. Conversion efficiencies at the 1% level are only possible with the greater than 100%
optical-to-THz photon conversion levels resulting from cascading effects. However, the spectral
30
broadening of the optical pump associated with cascading accentuates dispersive effects and strongly
limits further increase in conversion efficiency.
4.1 Theory
We first explore the impact of cascading effects using an effective 1-D spatial model. The electric fields
corresponding to the optical pulse at spatial co-ordinate z and various angular frequencies co are denoted
by E ,(w,z)=4,(a),z)e-jk(w)z . The THz electric field at spatial co-ordinate z and angular frequency
is denoted by Emz (C, z) = AHz(
, z)e-jk(fl)z.
Here, A,, (w, z) and Amz( , z) represent the envelope of
the optical and THz fields (henceforth referred to as field) respectively. The wave number for the THz
field is given by k(n) = fln(0) / c , where c is the speed of light in vacuum. Exact values of the THz
refractive index n(Q) are used to accurately model the material dispersion in the THz range. The wave
number for the optical field is given by (4-la).
(cv
k"()AD
(4-la)
k(1)
I
cosy
n
c
k "()AD =
+
2
-n g(0-)
_2
abcn(o-b)
(4-1b)
2
Here, the refractive index n(w) accounts for material dispersion. The factor Cosy appears in the
denominator of the first term in (4-la) to account for the tilted-pulse-front, where y is the pulse-front-tilt
angle. The second term k "(w)A , corresponds to GVD-AD. The expression for GVD-AD is given by (4Ib). The derivation of this expression is provided in Appendix A. In (4-lb), coo is the angular frequency
at which the pulse spectrum of the optical pulse is initially centered. The parameter ng (q) is the group
refractive index of the material at coo. Note, that higher order expansion terms can also be derived in (4-
1b) but it turns out that they minimally impact the results. When the tilt angle y=O, (4-la) reduces to the
regular expression for the wave number. Therefore, the developed model can be applied to both noncollinear and collinear OR geometries. The evolution of 4A Hz (,
dATHz (,
z)
dz
dz2
-n
2
2c k(Q)
a(f) ATHZ (0, z)
2
2 0(4-2)
(2 )(Z)f A 0P(0+ , Z)A
0
z) is given below in (4-2).
* (Cz)e-j[k(w+0)-k(w)-k(Q)]zd4
Here, the first term on the right hand side (RHS) corresponds to the THz loss due to absorption, which
may include background as well as FCA. The second term corresponds to the second order non-linear
polarization due to OR. Noticing this term, it is clear that OR is nothing but an aggregate of all possible
difference frequency generation (DFG) processes within the bandwidth of the optical pulse. It can be
seen that setting the exponential factor inside the integral on the RHS of (4-2) to zero, gives the well31
known phase matching condition n((i)cosr = ng(fty)
parameter
2
(z)
= 2 dffg(z)
is the effective second order non-linear susceptibility at the spatial co-
ordinate z. For the calculations in this chapter, Z
spatial variations in X
for OR using tilted-pulse-fronts. In (4-2), the
(z) is constant in space. However, by introducing
(z), the model can also be used to analyze quasi-phase-matching (QPM)
structures. Thus, the developed model can be used to simulate a number of devices based on OR. If only
(4-1)-(4-2) are solved, we would be working in the undepleted pump approximation.
In order to account for pump depletion and spectral re-shaping of the optical pump, we write the
corresponding equations for the evolution of AP(co, z) in (4-3). Equations (4-1)- (4-3) conserve energy
in the absence of any dissipative processes, which indicate that the theory is on solid ground. A proof of
this is provided in Appendix B.
dp
*~mz)
dz
-1w
2c 2 k(w)
-a, Ap (w, z)
2
z)
_ 2
2c2k(w) A
(z) Ap (w +z)ATHz
*(f)zdI
0
(4-3)
(z)I A0 (w- "z)A
)+
+F jeoojbn(ft) n2(z) Aop(Zt)j2 Ao,(zt) +F
{j
ofn(ft)2n2(Z) [jA 0p(z't-t)
@2
hr(t')
Aop(zt)
In (4-3), the first term corresponds to the absorption of the optical radiation. The second term is
analogous to the second term in (4-2), and represents the cascaded down-conversion of the optical
frequency due to THz generation. The third term represents the up-conversion of optical frequency
components due to sum frequency generation (SFG) between an optical frequency component at (co-)
and a THz field component at
. The fourth term corresponds to SPM. Here co is the permittivity of free
space and n2 (z) is the non-linear refractive index coefficient at spatial co-ordinate z. The symbol F is
used to denote the temporal Fourier transform. It can be seen that the effects of SPM are directly
proportional to the optical field intensity. The final term corresponds to SRS. Here hR(') represents the
Raman gain spectrum. The complex second order nonlinear polarization terms on the right hand side of
(4-2)-(4-3), were calculated rapidly by expressing them in terms of Fourier transforms. A 4& order RungeKutta (See Appendix C) method was used to numerically solve the above equations with a spatial
resolution of 10 pm. This spatial resolution was verified to achieve numerical convergence (See
Appendix C).
4.2 Validation of model: Comparison to experimentally measured optical
spectra
In order to validate the developed 1-D model, we simulate the transmitted spectrum of the optical pump
pulse at various levels of THz generation for the experimental parameters presented in [14]. The optical
pump pulse used in [14] was modeled by a transform limited 500 fs Gaussian pulse centered at 1031.8 nm
with a maximum fluence of 20 mJ/cm2 . The second order effective non-linear susceptibility Z& =dff
32
was assumed to be 360 pm/V [30]. A non-linear refractive index value of n 2
=
10-15 cm 2/W [31] was used
to account for SPM and SRS effects. The Raman gain spectrum was obtained from measurements in [32]
The refractive index and absorption coefficient data at optical and THz frequencies were obtained from
[33]. At 1 pm pump wavelengths, FCA of THz radiation occurs mainly due to free-carriers generated by
four-photon absorption of the optical radiation. Experimental fits [34] have estimated a four-photon
absorption coefficient of 10-7cm 5/GW 3 and a THz FCA cross section of 2x 10-21 cm 2 . Corresponding to
these values, even a 500fs pulse with peak intensity of 150 GW/cm 2 will result in a FCA coefficient of
only -0.1cm- 1 , which is much smaller than the background absorption coefficient of lithium niobate
(-10cm~ 1 at room temperature [33]). Therefore, for the peak intensities of 40 GW/cm 2 used in the
simulations below, FCA and four-photon absorption coefficients can be effectively set to zero with
negligible loss in accuracy. The crystal temperature was assumed to be 290K. The pulse-front-tilt angle
for optimal phase matching at 290K was y = 63.09'. A Fresnel reflection loss of -44% due to the high
refractive index of lithium niobate at THz frequencies is included in the calculations.
In the actual physical situation, different parts of the optical pump beam see different propagation
lengths due to the prism geometry of the crystal in OR using tilted-pulse-fronts. This situation can be
modeled accurately only with a 2-D spatial model, as will be done in Chapter 5. However, in the 1-D
model, an effective propagation length Leff is used. The simulated spectra of the transmitted optical pulses
are plotted in Fig.4-1 along with the experimental results. The conversion efficiency is tuned by changing
the pump fluence from zero to a maximum of 20 mJ/cm 2 . This mimics the rotation of polarization in the
actual experiment. The maximum conversion efficiency at room temperature was reported to be 1.15% in
[14]. We obtain a maximum value of 0.8% in our simulations, which is in good agreement with the
experimental results. The difference in results can be attributed to small uncertainties in material
parameters (;r&,n 2 ,aTH ) and experiments.
In particular, there is uncertainty in the absorption
coefficients below 0.9 THz [33].
Transmitted Optical Spectra
1
(a)-
Experiment [14]
-Theory
0.5-
0.4-
(b)
n
/2
0.4-
(C)
max
Theory =0.8%
0.2-
]
Experiment [14] =1.15%
1%10
1020
1030
1040 1050 1060
Wavelength(nm)
1070
1080
Figure 4-1 Comparison of experimental and simulated optical spectra for different amounts of generated THz. The
frequency down-shift and spectral broadening can be modeled only by the simultaneous solution of the THz and
optical fields. (a) No broadening or red-shift is observed when conversion efficiency I = 0 % , which implies small
33
effect of SPM. (b)There is large cascaded frequency down-shift and spectral broadening corresponding to a larger
amount of THz generation (c) Maximum frequency down-shift and spectral broadening is observed when
conversion efficiency is maximum at I,,,,.=0.8%. Conversion efficiency, amount of frequency down-shift and
spectral broadening are in good agreement with experiments [14]. The difference between theory and experiment
can be attributed to uncertainties in material parameters (4f,n 2 ,ar,) and experiments. The measured spectra
include spatial averaging effects which are not included in the calculations, which could explain their difference.
In Figs. 4-1(a)-4-l(c), both the simulated and experimental spectra show an amount of broadening and
red-shift commensurate with the amount of THz generated. This is strong evidence that the observed
broadening is a direct consequence of THz generation. To further reinforce this point, in Fig. 4-1(a), when
there is virtually no THz generation, there is also negligible broadening of the transmitted optical pump
pulse. This also indicates a relatively small impact of SPM and SRS effects in these experiments even at
.
the relatively large peak intensities of 40 GW/cm 2
In Figs. 4-1(b) and 4-1(c), the extent of broadening seen in experiments is well reproduced by the
simulations at half the maximum and maximum conversion efficiency, respectively. Here, in addition to
the red shift, a relatively small amount of blue-shift is also seen which also increases with increasing THz
generation. This effect is also observed in our calculations. An explanation for this is obtained by
inspecting the third term on the right hand side of (4-3). This term represents a blue-shift of the optical
pulse via THz plus optical SFG, which increases with the THz conversion. Consequently, there is an
increasing amount of blue-shift with increased THz generation, albeit to a much lesser extent than that of
the red-shift. The difference in the shape of the spectra between experiments and simulations is partly
due to the usage of a 1-D model. In the actual physical situation, due to the non-collinear propagation of
the optical and THz radiation, different parts of the optical beam will be broadened to different extents.
The final recorded power spectrum would then correspond to a spatial averaging of the spectral intensity
over the beam cross-section. Such spatial averaging has not been considered here.
The good match in conversion efficiency, frequency shifting, and spectral broadening between
simulation and experiments is evidence that even this simple 1-D model captures the essential physics of
the nonlinear THz generation process.
4.3 Limitation of conversion efficiency due to cascading effects
In 4.2, we established the accuracy of our model by obtaining good quantitative agreement with
experiments in conversion efficiency values and transmitted optical spectra. We now use the model to
show that cascading effects in conjunction with GVD-AD represent the strongest limitation to THz
generation for 1 pm pumping. We selectively switch-on and switch-off, various effects in our simulations
to study their relative importance.
In Fig. 4-2, we calculate the conversion efficiency as a function of the effective propagation length- Lff
for various cases. Material dispersion and THz absorption are considered in all cases. The different cases
are as follows :- (i) GVD-AD is considered. This corresponds to an undepleted pump approximation
where only Eqs. (4-1),(4-2) are used. There is no change in the total pump pulse energy or spectrum; (ii)
SPM, SRS and GVD-AD are considered. Equations (4-1) and (4-2) are used. Only the fourth and fifth
terms in (4-3) are employed. Here, the total pump pulse energy is conserved but spectral re-shaping of the
pump pulse spectrum occurs due to SPM; (iii) cascading effects are included but SPM, SRS and GVD-
AD are excluded. Equations (4-la), (4-2) and the first three terms of (4-3) are utilized. The second term of
(4-la) k"(O)AD is set to zero; (iv) cascading effects and GVD-AD are included but SPM and SRS are
34
excluded. Equations (4-1)- (4-3) are used but the fourth and fifth terms of (4-3) are ignored;(v) all effects
are simultaneously included (Eqs.(4-1)-(4-3)). A crystal temperature of 100K is assumed. This is because
the difference between various cases is best illustrated with cryogenic cooling rather than at room
temperature. The pump fluence is fixed at 20 mJ/cm 2 for all simulations as in Section 4.2. The material
parameters are the same as in Section 4.2, except for absorption and refractive index which are functions
of crystal temperature. Due to the change in refractive index with temperature, the pulse-front-tilt angle
for optimal phase matching is now set to y = 62.26'.
(a)
-FH=
FWHM
500
fS
() GVD-AD
(undepleted
pump
approx.)
8-
-
10
-----"
6-
iSPM,
GVD-AD
(iii)Cascading
Effiects
2
(iv)Cascading
GfectsGVD-A
0
1
L (m
2
4
10
5
6
(I)GVD-AD
(b) tFWHM=150fS
(undepleted pump
8-
6-
4.iiSPM
I....,.*.(ifl)Cascading
Effci
-
.--
-.
2-
iv)CascadIng Effects, GVD-6
(V)CaScading Effeo
GVD.AD, SPM
1
00
2
3
4
5
6
,
Figure 4-2: Conversion efficiencies as a function of effective length are calculated by switching on and switching off
various effects. Material dispersion and absorption are considered for all cases. The pump fluence is 20 mJ/cm 2
=2
360pm /V , n2=1 015 cm 2/W and crystal temperature is 1OOK.(a) Gaussian pulses with 500 fs pulse width at
FWHM with peak intensity of 40 GW/cm 2 are used. Cascading effects together with GVD-AD leads to the lowest
conversion efficiencies. The drop in conversion efficiency is attributed to the enhancement of phase mismatch
caused by dispersion due to the large spectral broadening caused by THz generation (See Figs. 4-1(b)-(c)).
However, since group velocity dispersion due to angular dispersion (GVD-AD) is more significant than group
velocity dispersion due to material dispersion at optical frequencies in lithium niobate, cascading effects in
conjunction with GVD-AD is the strongest limitation to THz generation. SPM effects are much less detrimental
since they cause relatively small broadening of the optical pump spectrum (see Fig. 4-1(a)). (b) Cascading effects
along with GVD-AD are most detrimental even for a 150 fs Gaussian pulse with three times larger peak intensity.
35
In Fig. 4-2 (a), we plot the simulation results for cases (i)-(v) for a transform limited Gaussian pulse
with a pulse width of 500 fs at FWHM. This corresponds to a peak intensity of 40 GW/cm 2. For case (i),
when only GVD-AD is considered, conversion efficiencies larger than 10% are possible, in agreement
with the results from [16]. For case (ii), when SPM and SRS effects are included, the conversion
efficiency peaks at about 6%. The reduction in maximum conversion efficiency is because of the
additional phase mismatch induced by the change in refractive index at optical frequencies due to SPM
effects. The increase in phase mismatch is evident in the reduction of the optimal effective length value
compared to case (i) to Lff= 6 mm. Beyond this distance, the coherent build up of THz energy reduces as
the optical and THz pulses slip out of phase.
In case (iii), when only cascading effects are considered along with material dispersion, the conversion
efficiency peaks at -3%. Here, the dispersive effects due to material dispersion are enhanced by the
broadened spectral bandwidth of the optical pump pulse and therefore the corresponding optimal effective
lengths are shorter compared to cases (i),(ii).This is demonstrated by a decrease of the optimal effective
length to -2.5 mm. This shows that cascading effects in combination with material dispersion are even
more detrimental to THz generation than SPM. This trend agrees with the experiments and simulations in
Fig. 4-1. Here, it was seen that the spectral broadening which occurs as a consequence of THz generation
was much larger than that due to SPM.
In case (iv), when both cascading effects and GVD-AD are considered, the peak conversion efficiency
is reduced to 1.8%. The optimal effective length reduces even further compared to case (iii) to -1.5 mm.
It can also be seen that the conversion efficiency degrades more rapidly with propagation length
compared to case (iii). These allude to the dominant effect of GVD-AD compared to material dispersion.
This is further verified by calculations of GVD-AD from (4-1b) (-15000 fs2/mm) which are significantly
larger in magnitude compared to group velocity dispersion due to material dispersion in lithium niobate at
optical frequencies (<1000 fs 2/mm). Finally, if all effects (SPM, GVD-AD, cascading effects) are
considered simultaneously in case (v), there is only a minor change compared to case (iv) in Fig. 4-2(a).
This confirms that the combination of spectral broadening caused by the generation of THz (cascading
effects) and angular dispersion are the most dominant limiting effects. A five-fold reduction in the
maximum achievable conversion efficiency in comparison to the 10% efficiency levels for similar pump
parameters [10] is observed.
In Fig. 4-2(b), we show the results for cases (i)-(v), using a transform limited Gaussian pulse with a
pulsewidth of 150 fs at FWHM, while keeping the pump fluence constant. This corresponds to a peak
intensity of -120 GW/cm 2 . At these higher intensities, the effect of SPM is seen to be more pronounced
than in Fig. 4-2(a). However, even here trends are consistent with Fig. 4-2(a) and cascading effects in
conjunction with GVD-AD impose the strongest limitation to conversion efficiency. The effective length
at which conversion efficiency peaks reduces even further compared to Fig. 4-2(a) for all cases. In case
(i), this is because of the larger effects of dispersion due to an increased spectral bandwidth compared to
Fig. 4-2(a). In cases (ii)-(iv), it is owed to a faster (with respect to length) broadening of the optical pump
pulse. This is because THz generation initially occurs at a faster rate because of the higher intensity.
However, since the generation of THz is necessarily accompanied by spectral broadening of the optical
pump pulse, coherent build-up of THz ceases at shorter distances.
36
4.4 A time-domain description
The rapid degradation of conversion efficiency with Leff can be understood more intuitively from a time
domain view point. In Fig. 4-3 (a)-(f), we present snapshots of the spatial intensity profiles of the optical
pump pulse at various times. The simulation parameters are the same as those used for Fig.4-2(a). In Figs.
4-3(a)-(c), only SPM and GVD-AD effects are considered. As the optical pulse propagates through the
nonlinear material, it is spectrally broadened due to SPM. Due to increasing spectral bandwidth with
propagation distance, the effects of dispersion become more adverse and the peak intensity of the pulse
drops as it broadens in time. The drop in peak intensity is accompanied with a drop in conversion
efficiency as expected. Comparing Fig. 4-3(a) to 4-3(c), the peak intensity drops by about 2.5 times over
the entire propagation distance. This effect is much more drastic when cascading effects and GVD-AD
are considered as seen in Figs. 4-3 (d)-(f). Here, the pulse shape is severely broadened and distorted in
time due to the much larger spectral broadening in comparison to SPM. The drop in peak intensity from
Fig.4-3 (d) to Fig. 4-3(f) is about 7 times. In addition, there is a severe break-up of the pulse in time. This
means different parts of the pump are generating THz radiation incoherently, which prevents further
coherent build-up of THz radiation. Therefore, the rapid degradation of conversion efficiency when
cascading effects and GVD-AD are considered can also be viewed as a consequence of the drop in peak
intensity and spatio-temporal break-up of the pulse.
0.8
SPM, GVD-AI
(b)
t=30ps
(a)
t=lops
(c)
t=50ps
S0.4
-0
0.8
Casc a ding Effects, G D-AD
(e)
(d)
t 5(f)
St=S0ps
t=30ps
t=lops
0.4
)
-0
4
8
Lj 6 nm)
4
8
Figure 4-3: Snapshots of spatial profiles of optical pump pulse at various times. (a)-(c) include only SPM effects
and dispersion (GVD-AD and material dispersion). Dispersion spreads the pulse in time, reducing intensity which
leads to saturation of conversion efficiency. (d)-(f) include only cascading effects and dispersion. The pulse is
dispersed much more rapidly compared to (a)-(c) due to the large spectral broadening.
37
4.5 Limitations to optical rectification for 800nm pumping
1.0
M
,
.0.6
',s-polarizatlon
040
''-
0.35
.
0.6
I
.
0.45
0.8
-
0.4
\p-polarization
%
0.2
0.30 - N
0.0.
I
0
820
Wavelength (nm)
840
801
802 803 804 805 806
Central Wavelength (nm)
807
Figure 4-4. (a) Experimental curves [29] indicating the relative broadening due to SPM and cascading effects for
pumping by 150fs Ti: Sapphire lasers centered at 800 nm wavelength. Even for these shorter wavelengths/pulse
durations, cascading effects dominate SPM due to the larger extent of broadening. (b) In addition to cascading
effects, three-photon absorption also plays a critical role in impeding THz generation. As the intensity gets higher,
three-photon absorption of the optical pump increases leading to a larger amount of free-carrier absorption of THz
radiation.
In this section, we provide a brief summary of experimental results [29] which followed up on our earlier
theoretical predictions [22] to confirm that cascading effects in conjunction with dispersive effects is the
strongest limitation to THz generation by optical rectification using tilted-pulse-fronts, even for pumping
by 800 nm pulses. In Fig.4-3(a), we present the results of spectral broadening with and without THz
generation, which confirms that spectral broadening in the absence of THz generation is significantly
smaller. This alludes once again to a smaller impact of SPM even for systems pumped by 150fs 800nm
pulses.
Fig. 4-3(b) shows the emitted THz energy as function of the central wavelength (2) of the pump pulses
for a fixed pulse duration T=298 fs and pump fluence of 7.6 mJ/cm 2 .The generated THz energy
monotonically increased with the central wavelength of the pump laser. When the central wavelength
increased from 801 nm to 806 nm, the THz energy was enhanced by -50%. This observation indicates
that the central wavelength of the pump laser has a strong influence on the THz generation in this
frequency range. We associate this dependence with 3PA. In fact, the direct band gap (Eg) of lithium
niobate is in the range of 4 eV at room temperature and a significant nonlinear absorption rate may
originate from 3-photon interband transitions, where the photon energy (hco) is-1.55 eV, even at moderate
intensities of up to a few tens of GW/cm 2 . For a given radiation intensity, the interband 3PA scales
approximately linearly with photon energy in the case 3ho > Eg [35], which is the case for 3PA in lithium
niobate with a radiation wavelength centered around 800 nm. This process limits the conversion
efficiency by significantly increasing the free carrier density, which then strongly absorbs THz radiation.
In previous work [16], FCA was cited as an important factor for saturation of THz energy. However, in
[18] it was suggested that FCA plays no significant role in the OR processes in lithium niobate and has no
effect at all in THz energy saturation. Moreover, SPM was recognized to be the only factor in the
saturation of THz energy. The results shown in Fig. 4-3(b) depict for the first time a strong correlation
between pump photon energy distribution on generated THz radiation and is evidence of the role played
by FCA arising from 3PA.
In summary, cascading effects broaden the spectral bandwidth which enhances dispersive effects and
compromises phase-matching. In lithium niobate, since GVD-AD due to angular dispersion is larger than
38
that due to material dispersion at optical frequencies, cascading effects in conjunction with GVD-AD
represent the strongest limitation to THz generation for 1 pm pumping. The inclusion of these effects
significantly reduces the maximum achievable conversion efficiency. The effective lengths at which the
conversion efficiency peaks reduce significantly and are on the order of -1 mm. For such short
propagation lengths, absorption plays a minor role in limiting conversion efficiency. This is an additional
reason why FCA is not expected to play a major role for pumping at 1 pm. For 800 nm pumping, in
addition to cascading effects and GVD-AD, three-photon absorption may lead to considerable FCA of
THz radiation [29].
39
5. The 2-D depleted model
In Chapter 2, we provided an overview of the physics at work in optical rectification. In Chapters 3-4, we
built up the complexity of the theory to a point that we now can dive into a comprehensive two
dimensional model which simultaneously accounts for (i) spatio-temporal distortions of the optical pump
pulse, (ii) cascading effects, (iii) SPM and SRS effects, (iv) material dispersion and absorption as well as
(v) nonlinear crystal geometry. An outline of the approach is as follows. The spatio-temporal distortions
imparted to the optical pump pulse by arbitrary imaging systems are modeled analytically via dispersive
ray pulse matrices. The emergent electric field of the optical pump pulse from the imaging system is used
as an initial condition to solve for THz and optical fields via a system of coupled nonlinear 2-D wave
equations. Spatial Fourier decomposition and appropriate co-ordinate transformations enable an efficient
solution of this coupled system of nonlinear equations. The use of spatial Fourier decomposition also
enables reflections of the THz radiation at the crystal boundary and subsequent propagation to be
modeled with relative ease. The developed model is applicable to the simulation of a variety of OR
systems with different imaging systems, crystal geometries and optical pump pulse formats.
We validate the model by comparisons of the optimal imaging configuration to analytic calculations,
which show good agreement. The impact of imaging errors on conversion efficiency is shown
quantitatively. It is seen that small perturbations to the optimal imaging configuration can result in
sizeable degradation of conversion efficiency and modification of THz radiation properties. We discuss
the meaning of effective interaction length in a 2-D geometry. It is seen that conversion efficiency drops
with beam size, which is a concern for scaling THz energies to the 10 mJ level by increasing the optical
pump pulse energy. We illustrate how the optimization of conversion efficiency in such setups is highly
dependent on the properties of the pump pulse. Furthermore, the optimization of conversion efficiency is
seen to negatively impact other properties such as the THz beam quality. This suggests that multiple
factors must be considered while designing such systems.
5.1 Theory
5.1.1 Propagation of optical pump fields through the optical setup
All THz generation systems based on nonlinear optical processes include some form of imaging
optics. The imaging system transforms the beam size and radius of curvature of the optical pump. The
variation of beam size changes the localized intensity whereas the finite beam size and curvature affect
phase matching by introducing a finite momentum (k,) distribution. Thus, output parameters such as
conversion efficiency, THz spectrum and THz beam propagation are linked to the imaging system
configuration. In non-collinear phase matching geometries, the role of the imaging system is even more
critical. For instance, for OR using tilted-pulse-fronts, imaging errors due to incorrect magnification or
grating angle affects phase matching by altering the pulse-front-tilt angle. Thus, a means to provide a
correlation between the THz spectrum, THz beam propagation and conversion efficiency to the imaging
system configuration would be very useful to quantify imaging errors and locate optimal imaging
conditions for a variety of different imaging systems.
In this section, we show how to account for the effects of the imaging system by employing dispersive ray
pulse matrices. This approach enables us to determine the modification of the optical pump field by an
40
arbitrary set of optical components. Since, the beam size of the optical pump is typically much larger than
the optical wavelength, paraxial approximations of ray-pulse matrix schemes are valid. Each spectral
component ETn(o,x,z) of the optical pump pulse at angular frequency o, with input beam centred at a
transverse position xi,(w) and propagation direction xj-(c) emerges with transverse position x, (o) and
propagation direction x,t(w) from the optical setup, just before entering the nonlinear crystal.
The dependence of position and propagation direction on frequency accounts for spatial chirp and angular
dispersion. The propagation direction here is not the physical direction but one normalized by the
refractive index of the medium [24], which makes it convenient to apply to systems with interfaces of
mismatched refractive index. The relationship between xi,(CO), xin(w) and x,, (c) , x,t(w)
is given by
(5-la). In (5-la), M(cp) is the overall ray pulse matrix of the imaging system and is given by the product
of the ray pulse matrices of individual optical components Mi(o) in reverse order of incidence as shown
in (5-1b). Here, propagation in free space is also considered to be an 'optical component'. The beam
positions and propagation directions after the it optical element, xw, (co) and x,, (o) can be obtained
using (5-1c). Note that the matrices are functions of frequency to incorporate dispersion.
xin (0) lXout
(CO,
0)
(5-la)
= xOUt(CO,0)
M(O) xin ())
(5-1b)
M(O) =M(0)..M()..M1(CO)
Xout, (')
~xin ()]
M1 (W)...M 2 ()MI(O)
(5-ic)
xin (CO) = xt,(CO)
The ray pulse matrix of the /h optical component M, (o) can be described by a 3x3 matrix for a single
transverse spatial dimension as shown in (5-2).
If instead, we consider both transverse spatial
dimensions, then we would have a 6x6 matrix instead.
OXout.
ax
x
OXin
ox
a
Ai(m)
Mi(C) = x
o)
-ax'
euxout
a0
0
Bi(oi) Ej(co)
=Ci(w) Di(o) Fi(w)
0
0
(5-2)
1
1
41
Equation (5-2) shows that the upper 2x2 matrix is nothing but the standard ABCD matrix for Gaussian
beams. However, in order to account for dispersion, there are two additional terms E and F which
correspond to the partial derivatives &bc/Bal, and ax,, 1/wa. These terms refer to the shift in output
beam position and output beam propagation direction in response to a shift in frequency. The subscripts i
indicate that they are different for various optical components. Typically, the E,(w) and F,(w) terms are
calculated upto the second order in (w -ct), which leads to an overall third order dependence on
frequency of terms. We provide expressions with up to a fourth order dependence on frequency for
common optical components in Appendix C. Note that the last row of M(co) is [0 0 1] for passive optical
elements as the source frequency does not change.
With the knowledge of the input and output beam positions and propagation directions, a Huygen's
integral can be used as shown in [36] to calculate the electric field of the optical pump after it has passed
through the imaging system. In this paper, a paraxial Gaussian beam is assumed for which the expression
for the electric field reduces to (5-3).
z')O
EOWt(w,xO,zO)=
0 -*20.0 #-j
u)t
E0e_
e0)
"
A(xo-x..,(o),Zo))2
2q.,,(w,zo)
-jkxo ,(WzO).-
e
(5-3a)
N
q0(mo) = k Y Lin,(c)
1
_
A(w, zo) + B(w, zo) / (w)in
k
0(w, zO) = -[xin(O)x
6(Ozo)=
(5-3b)
e-M(,zo)
aO,()
(5-3c)
GUt (w, zo)
(5-3d)
(O, zo)-x 0 t()xw (w,zo)]
k N
-
(5-3e)
J Fj(o)xt, (w)
2 j=1
In (5-3a), oti(w),cot(w) are the beam radii of the spectral component at c at the input and output of the
imaging system respectively. The pre-factor cj(o)/rOt(w) in (5-3a) represents the change in the optical
field intensity as a consequence of energy conservation. The second term in (5-3a), represents the
transverse variation of the beam along the xO co-ordinate axis, where qOut is the output q parameter
associated with Gaussian beams and is calculated with the usual A,B,C,D terms of the overall matrix M.
Notice that each spectral component is centered at a different beam position xoa (w,zo), which accounts
for spatial-chirp. Furthermore, this beam position is a function of space, i.e it changes as the optical pulse
propagates. The finite beam size in the transverse (xo) dimension results in a finite momentum (k,)
distribution, which in turn affects phase matching.
The third term, e-jx',(,Zo)xo represents angular
dispersion which also directly influences phase matching. Since, the value of
x
is mapped to the
42
configuration of the imaging system (grating angle, imaging distances etc), (5-3a) abets in quantifying the
impact of imaging errors on the properties of THz radiation (spectrum, propagation, conversion
are given by (5-3b)-(5-3e). The
efficiency). The various phase terms in (5-3a), #(co),(cv),(co)
Li represent
#o (c) term represents the overall phase delay due to propagation through the setup where
propagation distances through various optical components and nj(c) are the corresponding refractive
indices of each optical component at various optical frequencies. The
to #0 (w) due to a Gaussian beam. The remaining terms
02 (c) and
1 (co) term represents the correction
(co) account for all other dispersion
other than material dispersion and include the higher order phase terms. Thus using the ray pulse
matrices, one obtains an explicit expression for the electric 4"'(c,x0 ,z 0 ) field for every spectral
component of the optical pump pulse spectrum. This calculated Eot(cq,xo, zo ) value will be used as an
initial condition to solve the nonlinear coupled system of wave equations. This expression accounts for
spatial variations angular dispersion, spatial frequency-variations and spatial variations in pulsewidth.
Thus a one-to-one correspondence between the imaging system configuration and the properties of THz
radiation can be established.
5.1.2
Nonlinear polarization due to optical rectification
(z,x) = hsina,hcosa
J=0
n (09),
Initial conditions
BacR'
propagte
optical field
from Z 0
ETHz(
-
EP (co, x, -(L - h)sin a)
+ Zd)
iTHz Output
Field
-
ETHz(,x,-(L -h)sin a) = 0
x, h sin a
(2)=0
(z,x) = -(L-h)sina,-(L-h)cosa
Figure 5-1: 2-D computational space for solving coupled nonlinear wave equations for optical rectification.
2
Nonlinear crystal geometry is accounted for by delineating an appropriate distribution of )(x,z). Edges of the
distribution are smoothed out to avoid discontinuities. The refractive index is homogeneously distributed throughout
the computational space. The optical beam is centered at a distance h from the apex of the crystal which sets the
limits to the computational region. The optical field at the beginning of the lattice is calculated analytically using
dispersive ray pulse matrices.
,
In Section 5.1.1, we obtained the electric field of the optical pump pulse inside the crystal in the coordinate system (xo, zo) as shown in Fig.5-1. In this section, we show how to calculate the nonlinear
polarization terms which drive the optical and THz fields. The nonlinear crystal has an apex angle y
where y is designed to be approximately equal to the pulse-front-tilt angle of the optical pump pulse. This
design ensures that the generated THz radiation emerges approximately normal to the air-crystal interface
for maximum THz extraction. In order to treat crystal geometry and reflection at the crystal boundary
efficiently, it is prudent to work in transformed co-ordinates (x, z) where z is now along the direction of
the normal to the air-crystal interface and x is the corresponding transverse direction as shown in Fig.5-1
43
The origin of the (xz) co-ordinate system is taken to be the central position of the input optical pump
pulse and is located at a distance h from the crystal apex along the input crystal face of length L as shown
in Fig.5-1. Since all the THz radiation, emerges about a relatively small cone around the normal or
equivalently with a small spread in transverse momentum kx, the requirements for spatial resolution Ax for
Fourier decomposition in the (xz) co-ordinate system are much coarser compared to the case of working
in the (xo,zo) co-ordinate system. Thus in addition to mathematical convenience, the co-ordinate
transformation also alleviates computational cost. The transformation rules from (x, z) to (xozo) are
obtained straightforwardly via projection and are presented in (5-4).
xo =xcosr+zsinr
(5-4a)
zo =-xsiny+zcosy
(5-4b)
The electric field E0; (cx,zo)obtained from (5-3) is used as an initial condition to solving the nonlinear
system of coupled wave equations. We can use (5-4) in (5-3), to obtain the electric field of the optical
pump pulse in the transformed co-ordinates i.e. E,"t(,x,z). The value of E!2(c,x,z)
at
z = -(L - h) sin a serves as the initial condition for the evolution of wave equations described in Section
5.1.3. We first present the various nonlinear polarization terms which drive the wave equations in Section
5.1.3.
.
Equation (5-5) describes the nonlinear polarization term PTHz(n,xz)due to optical rectification (OR)
which drives the THz electric field E(O, x, z) at the angular frequency
00
PTHz(fx, Z)= O, z)f=I0(X).J
0
Ep (w+C,x, z)Eop
(5-5)
*(c,x, z)dw
In (5-5), E0p(o+Zx,z) corresponds to the electric field of the spectral component of the optical pump
pulse spectrum at angular frequency o and spatial location (xz). The nonlinear polarization term at each
spatial location (xz) in (5-5) can be seen to be an aggregate of all possible DFG processes between the
spectral component Eop(o+!4x,z) and Eop(cx,z). If one substitutes the expression for the electric field
from (5-3a). In (5-5), various effects which affect phase matching effects become evident. For instance,
because
of the
angular dispersion
term in (5-3a), a term
proportional
to
e
C
C
appears in (5-5) which enables us to quantify the effect of phase-
mismatch due to imaging errors. Similarly, a term proportional to e[C
2
q.t(w+f)
C 2qt,(w)
describes a spatial variation in the magnitude of PTHz (0, x, z) leading to spatial variation of the generated
THz frequency. This term also describes the effects of the finite radius of curvature of the optical pump
pulse.
In addition, substituting (5-3a) in (5-5) produces terms of the form
e
Awhich
gives the effect of phase mismatch due to various phase
44
accumulations through the optical setup. In (5-5),
( (x, z) is the effective second order nonlinear
susceptibility for OR at each spatial location and co is the free space permittivity. The spatial dependence
of the effective non-linear susceptibility is used to account for the geometry of the non-linear crystal as
shown in Fig.(5-1). In all locations represented by the shaded areas in Fig. (5-1),
(2
(x,z) 0 and
everywhere else it is equal to zero.
Similarly, each spectral component Eo,(,x,z) of the optical pump pulse spectrum is driven by a
nonlinear polarization term PJ,(co,x,z)described in (5-6).
-%,
(2, x, z)
z). ) E(w + , x, z)ETHZ * (fl,x, z)dco
(x,
= COnX)ff
0
+90 Zeff
~
(xP) 0 (o _ n,
0
x, z)ETHz (Q, x, z)dco
2k eoc2 F eoonw 22 (,Z
o
,x
)2 E (,XZ)(5-6)
2
2kzoeoc 2 F je60 mn(o )2n (x,z) [JE P(t-t,x,z)
2
2
0
,.(t')]Eo(tx,z)
The first term in (5-6) is the analogue of the right hand side of (5-5). It corresponds to the OR process and
signifies that an optical photon at angular frequency co is created by an aggregate of DFG processes
between optical photons at an angular frequency w+Q and THz photons at angular frequency Q. The
impact of the various terms from (5-3a) on the nonlinear THz polarization term PTHz(Q,x,z) is also seen
here. The second term corresponds to an aggregate of SFG processes between optical photons at angular
frequency w2- and THz photons at angular frequency Q. The third term in (5-6) represents the self-phase
modulation of the optical pump pulse. Here, Eop(t,x,z)is the time-domain electric field of the optical
pump pulse and F represents the Fourier transform between time and frequency domains. The intensity
dependent refractive index coefficient is n2(x,z) and has the same spatial profile as /) (x,z). The final term
models Stimulated Raman Scattering. This term is very similar to the SPM term but includes the effects
of a Raman gain lineshape given by hR (w)').
We thus see how impacts on THz generation arising due to spatial variations in phase and amplitude of
the optical electric fields in (5-3a) are captured by the nonlinear polarization terms.
5.1.3 Solving the 2-D non-linear wave equation using Fourier decomposition
In this section, we present our approach for solving the coupled system of nonlinear wave equations. The
nonlinear polarization terms defined in (5-5) and (5-6) will drive the corresponding THz and optical
fields. The nonlinear scalar wave equation for the evolution of the THz field ETH (ri,x, z) is presented in
(5-7).
V2ETHz(4,x,
z)+k2
z 4T'z( =x, z)= PTHz(A , Z)
60C
(5-7)
45
k(fl) = On(O) / c is the wave number at the THz angular frequency Q and n(Q) is the
In (5-7),
Similar to (5-7), one can also write the
corresponding refractive index at the THz frequency.
corresponding wave equation for the optical fields at various angular frequencies co in (5-8a).
v 2E0 (c,x,z)+k 2 ()E,(cox,z)=-2 JP(o,x,z)
(5-8a)
A brute force solution of (5-8a) can be cumbersome because of the rapidly oscillating phase terms due to
the large wave number of the optical field. In order to provide an efficient solution to the problem, we
define a solution of the form E0 (c,x,z)= 4p(wx,z)ejkxo(().xe-jko(w).z.
Here, kzo(w)and kzo(o) are
wave vector components in the x and z directions of the (xz) co-ordinate system, previously defined.
Making this substitution into (5-8a), we obtain (5-8b) for the evolution of the optical field.
2 A, (w,x,z) = -x2 Pjk(xz-xO*
-2jkzo 8AA0 (w,x,z) + 2jkAo ap(w,x, z)
az
EO2
&2
OJ.P(C
kzO(w)z
x~Z)e
(5-8b)
Equation (5-7) drives the THz field which in turn affects the optical field via the nonlinear polarization
term Pop(ox,z) defined in (5-6). However, the optical field directly influences the THz field via the
nonlinear polarization term PTHz(O,x, z) defined in (5-5). Thus, (5-7) and (5-8a) form a coupled system of
wave equations for THz and optical fields. An elegant solution to this system can be obtained via spatial
Fourier decomposition. This effectively breaks up (5-7) and (5-8b) into a system of coupled 1-D first
order differential equations which can be solved highly efficiently as will be shown below.
Applying Fourier transforms to both the left and right hand sides of (5-7) and we obtain (5-9a), which is
nothing but a 1-D differential equation in z for ETHz(rI,kx, z). The evolution of ETHz(f,k ,z) at each
and kx can be governed separately.
a
2(n_ x)ET2
2 k 2 )ETHZ(QXkZ,
Tz(,kz
Equation
(5-9a)
E(K,kx,z)
=
can be further simplified
A(C,kx,z)e-J( )z, where k.
=
z)=
'PTHz(Q,kx,z)
to (5-9b)
by assuming
(5-9a)
a
solution of the form
k 2 (9)-kx . The THz absorption coefficient at angular
frequency 0 is given by a(D).
z)
_
A(Q ,k, z)-
&z2
(2kz OC2)
p(2)Rk, z)ejk,().z
(5-9b)
X9
Similar to the spatial Fourier decomposition of (5-7), we Fourier transform both sides of (5-8b) to obtain
the corresponding 1-D differential equation in z for the optical fields in (5-1Oa).
aAoc*
kx, z)
0
caz
jkxo(cv)k
~
jkx2
A p(co, x, z) + 2k
o
kOC2
kzo(cv)
,x,
-
2
J Pop(co, k
2)
+kxO, z)eI
jkvr~
~~~
(5-1 Oa)
46
In (5-1Oa), the first term on the right hand side represents the oblique propagation of the optical field. The
second term corresponds to the quadratic phase associated with a Gaussian beam and the final term is the
nonlinear polarization term which includes all the cascading effects, SPM and SRS. Equations (5-9b) and
(5-10a) form a system of coupled I-D differential equations for each value of k,. However, the numerical
integration of (5-10a) requires a high spatial resolution because of the rapidly oscillating phase terms, i.e
the first and second term of (5-10a). To circumvent this computational challenge, (5-10a) can be reduced
to (5-1Ob) by noticing that A,, (co,kx , z) = #,,(o, k, z)e(Jko w)kx/ko()+kx/2k0)Z
e2
a 0 0 (co, k, z)
"
-
co
ko()k.
o
jk1
k *o(w) 2k,0)
Pop(o),k, -ko,z)e
(5-10b)
In (5-1Ob), we see how the phase matching condition can be altered by the oblique propagation and finite
size of the optical beam by examining the exponential factor. The dependence of the argument on kx
implies phase-matching condition will not be satisfied for all kx values. Alternatively, optical field
components at all values of kx do not participate in the THz generation process. This is analogous to
frequency dependent dispersion.
Equations (5-5), (5-6) and (5-9b) and (5-10b) are thus stepped through z, till we reach the end of the
nonlinear crystal at z = h sin a
5.1.4 Calculating transmission and propagation of THz radiation
Once (5-5),(5-6),(5-9b) and (5-10b) are solved iteratively until the end of the nonlinear crystal, we obtain
the THz electric field at 0 at the exit surface of the crystal. Next, we need to calculate the transmitted
THz electric field. For this purpose we employ the standard Fresnel reflection coefficients as a function of
the transverse momentum kx. For THz field polarizations perpendicular to the plane of Fig.5-1, the
Fresnel reflection coefficients are presented in (5-11). In [37], more specific reflection formulas for the
case when the group velocity of optical and THz are very disparate are provided. However, in the case of
tilted-pulse-fronts, this formula plays less of an important role as we are quite close to phase matching. So
we can use the regular Fresnel formulae.
n( C 2)
T(kx)=
22
2
X
2
(5-11)
2
k+2
X
In order to calculate the THz field at a distance zd away from the exit surface, we write a Huygen's
integral without any approximation as shown in (5-12a). This equation can be re-written in terms of an
Inverse Fourier Transform in (5-12b).
E(Qx,zd)= f A(Rk,hsina)T(k,)e AAokk zd
edkxxJcx
(5-12a)
47
E(f, x,z) = F 1
( A(f, kx, h sin a)T(k., )e- k0 -ki ) ZJ
(5-12b)
Thus using (5-1)-(5-12), we can model an arbitrary imaging system setup, combine it with a system of 2D nonlinear coupled wave equations to solve for THz and optical fields and then calculate the transmitted
THz radiation.
Despite the comprehensive modeling of the problem by (5-1)-(5-12), the computation is very efficient.
Equations (5-9b) and (5-10b) effectively form a l-D system of wave equations similar to [22] for each
value of k,. Therefore, the equations for various momentum values can be solved in parallel. The above
system of equations was integrated using a 40' order Runge-Kutta method. The evaluation of Fourier
transforms was accelerated by GPU parallelization (See Appendix D for further details).
5.2
Validation of the model
In this section, the developed model is validated against analytic theory and experiments. Simulations
assume a simple grating and lens setup to create the tilted-pulse-fronts, in line with the experiments in
[14], [21]. The full-width at half maximum (FWHM) pulse duration was assumed to be 0.5ps with a
fluence of 20 mJ/cm 2, and a peak intensity of 40 GW/cm 2 was assumed as in [22]. The second order
effective nonlinear coefficient was assumed to be XeP = 360 pm/V and the intensity dependent refractive
index coefficient was n 2=10-" cm2/W. which is in the range of values specified by [30]. As a first step of
verifying the accuracy of the developed model, we simulate the conversion efficiency as a function of
the grating incidence angle (6), grating to lens (s1 ) and lens to crystal (s2) distances. This is plotted in Fig.
5-2(a). It can be seen that an optimum conversion efficiency of 0.7% is obtained at s,=60.89cm,
s2=36.84cm and 9i=46.50. Analytic calculations for the optimal imaging conditions [17] are presented in
Fig.5-2(b). Here, for an optimum pulse-front-tilt angle y-63 0 , the corresponding imaging distances very
closely match the simulation results, i.e. sl=60.19cm, s 2=37.23cm and 9j=46.5o. The experimentally
obtained conversion efficiency (1.15%) is higher than the theoretical prediction, which is partially owed
to uncertainties in certain material parameters [33]. This agreement in optimal imaging conditions is a
strong validation of the developed model. In addition, the spatially averaged THz and optical spectra at
the output facet of the crystal are presented in Figs. 5-2(c) and 5-2(d) respectively. The THz spectrum is
centred at -0.45 THz and is in close agreement with experimental measurements [21]. The optical
spectrum is however broadened to a larger extent than the experimental results. One possible explanation
for this is that the experimental measurements potentially suffer from clipping of the more divergent
higher optical frequency components.
48
SimuLation of Optimal Imaging Conditions
52
0.7
61.5
0.6
65
Theory :Optimal Imaging conditions
1
s
60
61
E
60.5
350
10.4.0
0.3
36
y=62.55
i1 Vs0.2
.89CMe
...
36.5
0.1
M
37
37.5
s 2(cm)
60
55
55
0.4
60
59.5
=60.19Cm
65
40 -
S2=37.23cm
0
38
55
60
65
40
7P
(b)
(a)
THz Spectrum
Otical Spectra
1 -'
(c)
(d)
1-
0.8
0.
N
o.66-Theory
-Experiment
0.4-
0.
4-
0.2
0. 2
0
0.5
1
1.5
2
980 1000 1020 1040 1060 1080 1100
Frequency (THz)
Wavelength(nm)
Figure 5-2(a): Simulation of conversion efficiency as a function of imaging conditions. The surface plot shows the
conversion efficiency versus the imaging distances s1, s2 which are the lens to grating and lens to crystal distances
respectively. The inset shows conversion efficiency versus incidence angle to diffraction grating. (b) Theoretical
calculations of optimal imaging conditions for various pulse-front-tilt angles. For a pulse-front-tilt angle of -63
degrees, the imaging conditions are in close agreement with the simulation results, validating the accuracy of the
presented model. (c) The experimental and theoretically calculated spectra are presented. (d) The output optical
spectrum is presented along with experimental results.
5.3 Understanding effective length in two dimensions
In the remainder of this paper, the procedures to optimize the conversion efficiency and the
accompanying trade-offs are discussed. As a starting point for this discussion, it is useful to understand
what the effective length in a 2-D geometry is since this is usually the parameter used to describe the
previous 1 -D models. In general, two factors determine what the optimal effective propagation length Leff
is. These include absorption and dispersion. A longer effective length leads to more absorption and
dispersion. A short effective length would mean less of both but it would also translate to a smaller
amount of THz generation. Therefore, there exists an optimum value where the amount of THz generation
is sufficiently large but absorption and dispersion are sufficiently small.
49
In a 2-D non-collinear geometry, two parameters directly influence absorption and dispersion. These
include the beam radius o and the beam position h. These parameters were introduced previously but are
presented for convenience once again in Fig. 5-3(a). The THz and optical beams propagate noncollinearly. Since, THz is generated only in the region where there is optical fluence, a larger value of h
would mean that the overlap of the THz and optical beams would reduce and the THz would propagate
over a longer absorptive region (i.e without optical fluence). Similarly, the optical beam would propagate
longer distances which would mean they would suffer greater spectral broadening due to cascading
effects and SPM. This in turn would make dispersive effects more acute. Therefore, one would expect
that a larger value of h is necessarily more detrimental to THz generation. The exception to this would be
when the optical pump beam is clipped or equivalently is not completely contained within the nonlinear
region. Similarly, a larger value of beam radius a would lead to a smaller overlap and a larger amount of
absorption. Also a larger value of a would lead to a greater amount of overlap, which would in turn lead
to a greater amount of cascading and would accentuate dispersive effects further. These various effects
are shown in Fig. 5-3(b)-(d). In Fig. 5-3(b), a beam with radius 2.5 mm, is placed 1.5 mm from the crystal
apex. It can be seen how the THz and optical beams have good overlap which reduces absorptive effects
and leads to a high conversion efficiency. In Fig. 5-3(c), the same beam is displaced further down the
crystal. One sees that more THz absorption occurs leading to a large drop in conversion efficiency. In
Fig. 5-3(d), a large beam size of 10mm radius is used which is very close to clipping the crystal. We see
that only a small portion of the optical pump beam cross-section produces THz radiation. This is because,
after initial THz generation, subsequent parts of the beam are spectrally broadened to an extent which
prevents them from further generating THz. This has a powerful impact on the scaling of THz energies
with large pump energies. If, the intensity is maintained constant and the beam size is enlarged, the
conversion efficiency is not maintained as per earlier expectations due to the phase mismatch caused by
the broadened optical spectrum. This places important limitations on the scaling of THz energies using
large optical pump beam radii. An approach to circumvent this is to use an elliptically shaped pump beam
with major axis perpendicular to the plane of the tilted-pulse-front (i.e in the y direction). Thus for a given
intensity, both the beam position and beam radius have to be optimized in order to optimize conversion
efficiency.
50
0.3%
0.32%
0.65%
4
2
2
2
.2
S1-2
1-
mx-4
P- 0 P MP-2
0
Z(mM)
(a)
(b)
1
2
.1
0
1
-82 1
(m)
Z(m)
(c)
(d)
Figure 5-3(a). Effective length in two dimensions. The beam position h and beam position a affect the amount of
absorption and dispersion and affect the effective length in 2-D. Absorption is larger for larger h since there is less
overlap with the optical beam, whereas a is proportional to overlap and therefore reduces absorption. An increase in
h or a increase the effective propagation distance of the optical beam and therefore increase dispersive effects.
Therefore for (b) small h and a, there is minimal absorption (c) for large h and small a, absorption increases and
conversion efficiency drops (d) for large a, only small portions of the beam are involved in THz generation, leading
to an overall drop in conversion efficiency. This has important implications on the scaling of THz energies, by
merely scaling beam size.
5.4 Considerations for optimizing conversion efficiency
Now that we have established how to think in terms of dispersion and absorption in terms of o and h, it is
useful to look at what factors influence absorption and dispersion. The main source of dispersion is
bandwidth of the optical pump pulse, which in turn is dependent predominantly on spectral broadening by
cascading effects. This broadening increases with both initial optical pump intensity as well as initial
bandwidth. Thus, when either optical pump intensity or bandwidth are changed, the effect of dispersion
changes which in turn affects the optimal effective lengths or equivalently, the corresponding optimal
values of beam radius and beam position.
Changes in absorption and dispersion will change the optimal effective length which in turn affects the
optimal h and optimal o-. However, dispersion is affected by both the intensity as well as the bandwidth of
the input optical pulse and therefore for different intensities/bandwidth, the setup has to be optimized
differently. Equivalently, the optimal intensities/ bandwidths are different for different beam sizes and
different temperatures. This makes optimization of the THz generation system a complex problem.
Suppose the peak intensity is increased, this would lead to a faster rate of THz generation, which would
in turn cause spectral broadening to occur faster. Therefore, the effects of dispersion would be
accentuated over a shorter distance compared to before and the optimal effective length would reduce.
This can be achieved by a reduction of beam position or beam size, while keeping the peak intensity
constant.
51
In Fig.5-4, we plot the conversion efficiency as a function of the peak intensity of the optical pump pulse
for different beam positions h. It can be seen that the shape of the curves is different for different beam
positions. Each curve shows a different saturation intensity or conversely, there is an optimal beam
position for a given peak intensity value. This could be one reason why, various experimental report very
different saturation curves. Thus, in optimizing conversion efficiency, a careful optimization of the
experimental setup is required.
r
0.75
-+*-h= 2.5mm
7--+-h=
2mm t6
.6.-+-h= 3mmC
0.65
;N-8i
10
'u0.4
-~ 0.6
-15
-10
-5
0
0.5
5
x 10
x(m)
0.55
X 10
;.I8
10
0.45
'0.6
44
0.4 ..
S
C&0.2
5
10
15
J/cm2
20
20ence
-20
-10
1
0
10
Figure 5-4 (a) Conversion efficiency saturation curves for different beam positions h. The saturation energy/fluence
is dependent on the beam position. (b) While conversion efficiency may increase with increasing intensity, other
detrimental effects such as significant spatial-chirp of the THz spectrum also occur. Therefore, optimization of such
systems must consider other aspects of THz radiation while optimizing the setup.
In the previous two sections, we have illustrated the concept of effective length in 2-D and how it
translates to the complexity of optimizing the experiment. The next question, we pose as a conclusion to
this thesis is whether optimizing conversion efficiency should be the only priority in the construction of
such setups. Evidence for this is shown in Figs. 5-4(b) and 5-4(c). It is seen that while the intensity does
increase conversion efficiency, it also introduces a large spatial-chirp in the THz spectrum. Such a spatial
chirp is detrimental to beam transport, focusing and therefore would be perhaps less useful than one with
marginally lower conversion efficiency but superior beam characteristics.
52
6. Conclusion and future work
In the pursuit of optical-to-THz conversion efficiencies on the order of 10%, pulse energies of - 10 mJ
and pulse durations in the range of a few ps to few hundred picoseconds, we turned to the approach of
optical rectification via tilted-pulse-fronts in lithium niobate. While earlier theory predicted conversion
efficiencies as high as 10%, experimental results have been signficantly smaller. This led to a reformulation of the theory in greater detail. The main improvement was the consideration of the coupled
interaction between the optical pump and generated THz radiation in two spatial dimensions. This
revealed that a simultaneous broadening of the optical pump pulse in momentum and frequency occurs as
a natural consequence of THz generation. This leads to a rapid spatio-temporal break-up of the optical
pump pulse. The implications of this new physical insight are as follows.
(i)
(ii)
(iii)
(iv)
Since THz generation necessarily leads to this spatio-temporal break-up, the optical
rectification process is inherently a self-limiting process.
The effective length of THz generation reduces to a -few mm which drastically reduces the
predicted conversion efficiency.
The reduction of the effective length also implies that scaling of THz energy to very large
beam sizes will lead to a drop in conversion efficiency, which had not been anticipated
previously. This can be circumvented by the use of an elliptically shaped beam, with major
axis perpendicular to the plane in which the pulse front is tilted.
The spatio-temporal break-up of the optical pulse causes large spatio-temporal distortions in
the THz properties. This affects the beam transport and focusing properties of the THz pulse.
The above insights into THz generation pose important questions which shall be the emphasis of future
work. Firstly, it would be important to find out what the maximum achievable conversion efficiency in
lithium niobate is, under any arbitrary circumstance. Next, methods to compensate for detrimental
dispersive effects have to be considered. Operation at longer pump-wavelengths to reduce spectral
broadening is yet another important consideration. Then there is the combination of these various
approaches in tandem. The thesis thus puts the solution to the problem on the right path, to deviate from
existing approaches and to re-think the strategy to push the frontier of high-field THz generation.
53
7. Appendix A: Tilted-pulse-fronts
First, one can show how an angularly dispersed pulse results in a tilted-pulse-front. Note that the converse
is not necessarily true. A tilted-pulse-front can also result from a spatially chirped pulse. A detailed
discussion of various spatio-temporal distortions and their correlation to pulse-front-tilts is provided in
[23]. Here, the endeavor is purely illustrative and not meant to be exhaustive.
_ 2
r2(W_-j)2
)-)]z,-jkfl._Ry
ee -j+v,
E(a), x, z) = Ee
(A-1)
2 )2
E(t, x, z) = ejt
J
Eoe
=ejtejoz
f
e
4
2
rW
*e~jk*w"Me~O''
2e
2
x
e
Eoe~
2
-kofda'
a2ejt-vg
(t-Vg IZ-kopX)
-> E(t, x, z) = Eoe
ejw''do'
2
2
2
X
e
a2
e
,-4z)
Inspecting (A-1), we see that the all points with the same value of t -v z -k px have the same intensity.
This describes an inclined or tilted intensity profile. Second, the phase velocity is still z directed,
indicating that the phase fronts are still perpendicular to the propagation direction. Based on our definition
of the angle y in Fig. 1-1, the angle y is determined by the derivative of z with respect to x.
dz
.dO
Therefore,- = ko,8vg = tan y. However, 8 is nothing but the angular dispersion dx
dcoa
have the following relationship between the pulse-front-tilt angle and angular dispersion.
-Au
n(4) dO
6
g (A.~t)
-
.
Therefore, we
Rl
-=A) an y(A-2)
d%
Using, (A-2) we can derive the group-velocity-dispersion (GVD-AD) due to angular dispersion used in
Chapter 4. Here, f(Jo) is the propagation vector, propagating at an angle 0~0. The corresponding
second derivative can then be explicitly calculated, and by substituting (A-2), (A-3) is obtained.
f/(co) = k(co)cosOa, +k(c)sin0a^
2A
d213(co
m) dO 2(do
= k( ) dc) = -n(k) d)
d2
dA
(dco
dCO2
d'8(o)
dCO2
=ft
ng()tan
n(4 )co
27rc
(A-3)
27
54
8. Appendix B: Energy conservation in optical rectification
The one-dimensional equations for THz generation by optical rectification are described below in (B-1) to
(B-2).
*
z
A 0, (
-j
n)A T Z * (
0)+
' 2A.,(k-)
)e-[kQ( +n)k(- Wk( w)zdM -
e-,')
k( 0 )
-k(Q)-k()Izd n
(B-1)
(B-2)k~wfl
2 2
,,(
=
~~iATHZ(~I)
0( + Q2)A0,, *
2k(
dz
(B-2)
>Id
*
))
The total Poynting vector is written below in (B-3).
S,,(z) =
2
) A,
(Wz)1
0 Apo
2
dco+l k(2) I
2 0 92p"
z
(B-3)
dQ
To check for energy conservation, the spatial derivative of the Poynting vector must vanish in the absence
of any energy dissipative mechanisms. The spatial derivative is written in (B-4).
)
dS,,,(z)I k(co)
(,z
Ap
f
2
dz
0 'WP
0O-
+
k(
2 o QPto Ldz
dp*(+, z))
*(oz)
+A
dz
(, z) d)*+ATHZ
d,(,Z)
dz
d
(B-4)
dA(Cz dzM
dz
I
* (0,z)
Substituting (B-1) ad (B-2) in (B-4), we have the following expressions in (B-5) and (B-6) for the spatial
derivatives of the optical and THz radiation respectively. Here cc is the complex conjugate of the
presented expression.
0000
(2)
f-j 2
A,*()A 0,,(c + 2)A,
0o
+f f-iu
000
(A,*(W)A
0, (C
QA
* (Q)e-j'0+(w*)-k(n~k(w)Idrd
p
+cc
(B-5)
--[k(-I)+k(fI)"
j)(
k(w)zd(dd
+cc
2)
2po
S
-i 2p
4
*
)A,*(
(flA,(i+
-t(*-(1"*
'd+dC~C
(B-6)
Simple algebraic manipulation of the second term of (B-5), i.e, making the transformation of variables
from co to w+Q, allows for re-casting (B-5) as (B-7).
fj
2
A *(ov)A,(o) + )Am
* (Qe-ilk(
**)-k(O)k(wzd~d+cc
00
00
+ffj
N of
(
(B-7)
(+2)*+
* A, (cv + 2)A0,
(n)e~j[k(+
2 O A (w i eqa t zeo an
n)-])-k(*1dadw+cc
Note that the sum of (B-6) and (B-7) is equal to zero and hence the conservation of energy is established.
55
9. Appendix C: Numerical convergence of depleted model
A 4t order Runge-Kutta method was used for numerical integration. Euler integration approaches do not
work sufficiently well for propagation over long distances or for the 2-D implementations. For 2-D
geometries, the problem is relatively stiff and requires very fine spatial and temporal resolution, due to
simultaneous broadening in spectral and momentum domains. A number of n(n+J) Runge-Kutta methods
were tried, i.e. 4(5), 5(6), 6(7) and 7(8). However, the 4t order method appears to work better even
compared to the 7(8) approaches. Here we present the numerical algorithm for the 4th order Runge-Kutta
implementation in (C-1) through (C-5).
THz(op)
Q(cw),k,,1
HT,
TH(op)
I,k
k,
(C-1)
z)
(,k,,1
= Az
'THz(op)
(),kx,3
THz(op)
1
&
fTHZ( 0 P)
ATHz+
Qlkw,,2
Q~)kzA
(w),k,,z
Hz
,k,2
A
lop
,z
OP
THZ(OP)
~2
z
,k,,3
,kz
A
Q(w),k,,
~
1
6Hz(op)
THz(op)
(C-2)
A 0 IZ + lop 2
2
=AZ.THzI
THz(f(op)
THz
w,k,,l
(C-4)
,k,,3
,z
Q2(o),kx ,3
)k,2
THz(op)
(C-3)
Iz
2
THz(op)
fQ(a),k,,4
THz(op)
(C-5)
For the simulation parameters described in Chapter 4, we propagate over a distance of 4mm and notice
the convergence of results at a relatively sparse resolution of 10 pm. The THz and optical spectra and
phases are plotted below in Fig. 9-1.
6X 101
so
-dz=0.01mM
-- dz=0.005mm
- dz=0.001mm
-- dz=O.O005mm
0
-50
-
5
4-
W-100
M
rq
.- 100
E4-200
3
2
-250
1
-300
A
-%aau
5
10
Frequency (THz)
15
20
0
1
2
Frequency (THz)
3
56
r10
1001
5
4
0
-so
0.
2
-100.
~nL
-1-1
-050
k
I
1000
1050
Wavelength(nm)
1100
1150
0
1000
1050
1100
1150
Wavelength (nm)
Figure 9-1. Numerical convergence of simulation for (a) THz spectral phase. (b) THz spectral magnitude.(c) Optical
phase and (d) Optical spectrum for propagation over a distance of 4mm in Lithium Niobate.
57
10. Appendix D: Notes on 2-D implementation
Vectorizing calculations
MATLAB is not well suited in handling for-loops. Since we have a number of variables which are
functions of frequency/time, x/momentum (k,) and z, looping structures must be minimized. Since the
point of spatial Fourier decomposition is to enable a parallel solution, upto 2 dimensions can be solved in
parallel and only a single looping structure (looping through z) would be required. Therefore all
operations in o and kx were vectorized.
Representing Nonlinear Polarization terms as Fourier Transforms
The nonlinear polarization term driving the THz field can be written as follows. Note that, here we switch
to an integral from -co to oo instead of the previous definitions where it was a summation from 0 to co.
This lends itself to being calculated via Fourier transforms as will be shown below.
P(Qx, z)=
(D-1)
E(w+ Q,x, z)E, *(c,x, z)d c
z) foE(x,
Algebraic manipulation leads to (D-1) being written as (D-2).
P(,x, z) =e
(x,z) f E., (t, x,
E * (o,x, z)dw
-W-00
00
= eZs (x,z)f E,(t,x,z)
0
{f e"'E(m,
x,
00(D-2)
*e-i'dt
Finally, (D-2) can be expressed in terms of Fast-Fourier-Transforms (FFT) and Inverse-Fast-FourierTransforms (IFFT) below in (D-3). Only the latter half of the calculated array will be used since the first
half does not correspond to a physical frequency.
P(, x, z) =
CoX) (x, z)
Ec(t,x, z)E,
*(t,x, z)e-dt
= EoegfFFT( IFFT[E,(co,x,z)] 12)
(D-3)
Similarly, the nonlinear polarization term driving is written in (D-4) and simplified to (D-5) on the same
lines as (D-1) to (D-3). Once again, the physical THz frequencies only extend from 0 to oo . However, for
ease of computation in terms of Fourier Transforms, an integral from -oo to oo is considered. It follows
then that the THz field for negative frequencies is defined to be the complex conjugate of the positive
frequencies, i.e. E(-f) = E * (a).
P(, x, z) = eo eZf (x, z)f E(w - 0, x, z)ETH,(0, x, z)d Q + eO) Eop(w+,
0
=
f
f
0
x, z)EH
*(9, x, z)d 0
(D-4)
EP(w +n,x, z)ErH *(G,x,z)d(
58
P(W,
x, z) = 60
(x, z)FFT{IFFT[E,
(c, x, z)].(IFFT[E
,(K,
x, z)]*)}
(D-5)
The above implementation significantly speeds up computations of the nonlinear polarization terms. The
validity of calculating transforms via this approach is seen in their convergence with explicit calculations
of (D-1) and (D-4) in Fig.10-1.
X1010
-Brute
-1-d
3-
force
-2-d
2.5
1.510.5
1020
1030
1025
1035
1040
1045
Wavelength(nm)
Figure 10-1 Convergence of optical spectra calculations via brute-force (D-1) and (D-4) and Fourier transform
approaches (D-3) and (D-5). The propagation length was 2mm. Beyond this propagation length, the brute-force
approach is numerically too cumbersome to evaluate.
Notes on Fourier Transforms
In the actual MATLAB implementation, applying FFTSHIFT twice does not undo the action of the first
operation in all cases. Since centering the spectrum is necessary for any operations on the spectrum but
subsequent operation may require reverting to the time domain and vice-versa, the following definitions
for the Fourier transform were used to ensure seamless universal operation.
FFTcustom = FFTSHIFT{FFT{IFFTSHIFT{}}}
IFFTcustom = FFTSHIFT{IFFT{IFFTSHIFT{} }}
Notice that the two operations are inverses of each other.
Parallelization and GPU implementation
GPU operation significantly speeds up the computations of Fourier transforms. A 4x improvement in
speed is obtained with appropriate GPU implementation. One shortcoming is that not all the data may be
easily retrieved due to the limited size of the GPU memory. One approach to circumvent this issue is to
periodically extract the information from the GPU memory. However, this incurs a communication
overhead. Therefore, there is an optimal number of data extraction processes that need to be carried out.
Dispersive Ray Pulse Matrices for common elements
Propagation through material of refractive index n(co) and length L
59
L /n(co)
0
1
0
0
-0
1_
I
0
(D-6)
Diffraction grating
The 6x6 matrix for the diffraction grating is presented below. The derivation follows straight forwardly
(c)
from the grating equation, i.e sin 9, + sin 9
F- =
2;rc
~A
0
0
0
D
F(co)
L
0
1
, where p is the spacing between grating lines.
(D-7)
-2;rcnC%2 CCo
2 COS
C I-
A
-_
(D-8)
(D-9)
d=
cos9 1
nm)
D= cos (1- tanod F
cos9o
n(coo)
)
(D-10)
F,=F01I-(coC
CO
+FItan
2)
odF
22
F = Fo
--o
(
CO
+I tanOF + Fo2
)
2
(D-11)
6
F(o) = F + F, + F2
60
11.
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63